Convolutional codes from units in matrix and group rings

Con v olutional co des from units in matrix and group rings T ed Hurley Abstract A general meth od for constructing conv olutional co des from units in Laurent series o ver matrix rings is presented. Using group rings as matrix rings, this forms a basis for in-depth exploration of conv olutional co des from group ring enco dings, wherein the ring in the group ring is itself a group ring. The metho d is u se d to algebraically construct series of con volutional cod es . Algebraic meth ods are used to compute free distances and to construct conv olutional co des to prescrib ed distances. 1 In tro duction Metho ds are presented for constructing conv olutional co des using units in Laure nt series of finite supp ort ov er matrix rings. By consider ing group rings a s ma trix rings, con volutional codes a r e constr uc ted fro m units in La uren t series over group rings; these may b e co ns idered as group rings over group rings. Thus conv olutional co des are constructed b y considering a group ring RG where the ring R is itself a group ring. The metho ds are based on the gener al method in [3] fo r constructing unit-derive d co des from gro up rings wher e now the ring of the group ring is a group ring and the group of the gro up r ing may b e an infinite gr oup suc h as the infinite cyclic gr o up. F or genera l infor mation on group ring s and r e lated algebra see [9 ]. Using these algebraic metho ds, the range of co n volutional co des av ailable is expanded a nd ser ies of conv olutional co des are derived. F ree distances and co des to a pres cribed free distances may also b e derived. Indeed many of the existing conv olutiona l co des can b e obta ined in the manner of this pap er. The pap er [8] is an often quo ted sourc e of informa tio n on conv olutional co des wherein is mentioned the lack of algebraic metho ds for constructing conv olutional co des; and that many of the existing o nes hav e bee n found by computer se a rc h and are of necessity of relatively short memories . The methods ar e fairly gener al and use prop erties of group rings and their em b edding into matrix rings. Zero-div isors and units in g roup rings enables the c o nstruction of units in certain p olynomial rings and/or group r ings ov er these gro up rings from which conv olutional co des can b e cons tr ucted. Prop erties of the con volutional co des c an b e studied and der ived from pr operties of group rings. In man y instances the free distances can b e calculated algebraically and conv olutional co des to a sp ecified free distance, as for exa mple in Theo rem 7.3 or Theor em 14.1 be low, can b e constructed. The following are some of the applica tions of the gener al method and these in themselves constitute new metho ds for constructing co n v olutio na l co des: • The co nstruction of series o f binary (2 , 1) co n volutional codes and calculation of their free distances using the group ring ( F C 2 ) C ∞ where F is a field of characteristic 2; • Given a linear cyclic co de C with d = min( d 1 , d 2 ) where d 1 is the minimum distance of C and d 2 is the distance of the dual of C , the gener a tor p olynomial f of C is mimick ed in R C ∞ to construct conv olutional (2 , 1 ) codes of minimum free distance d + 2; • The cons truction of rate 3 4 and higher ra te co n v olutio na l co des with prescrib ed minim um distance; • The co nstruction of conv olutiona l codes ov er a field F of characteristic p for any prime p using nilpo ten t elements in the field F G where G is a gro up whose order is divisible by p ; • The construction of Hamming typ e c on volutional co des and ca lculating their free distances ; the construction of Hamming-type co n volutional codes to a desired minim um free distanc e ; 1 • The construction of co n volutional co des using idemp oten ts in gro up rings. These are particula rly used in c a ses where the c haracteris tic of the field do es not divide the or der of the group; char acters of gro ups a nd char acter t a bles come into play in construc ting these conv olutional co des. 1.1 Algebraic Description of Conv olutional Codes Background on gener a l a lgebra and group r ings ma y b e obtained in [9]. F or any r ing R , R [ z ] denotes the p olynomial ring with coefficients from R and R r × n denotes the ring of r × n matrice s with co efficient s from R . R n is use d to denote R 1 × n and thus R n = { ( r 1 , r 2 , . . . , r n ) : r i ∈ R } . It is easy to verify that R r × n [ z ] ∼ = R [ z ] r × n . R [ z , z − 1 ] is used to denote the set of Laurent series of finite support in z with co efficien ts from R . Finite su pp ort mea ns that only a finite num ber of the co efficients are non-zer o. It is clear that R [ z , z − 1 ] ∼ = RC ∞ , wher e C ∞ denotes the infinite cyclic group. (Elemen ts in g roup r ings hav e finite suppo rt.) Note also the r e lationship b et ween R [ z ] and RC ∞ – R [ x ] ∼ = T where T denotes the algebr a of non- ne gative elements , i.e . the algebr a o f elemen ts w = ∞ X i =0 α i g i , in R C ∞ . If F is a n in tegral domain then F [ z ] has no ze ro-divisors and only trivial units – the units of F [ z ] are the units of F . See [8] and/or [1] for basic information on conv olutional codes and algebra ic descriptions are describ ed therein. The (eq uiv alent) algebraic descr ipt ion given in [2] is ex tr emely useful and is given below. A con volutional co de C of length n and dimensio n k is a direct summand of F [ z ] n of rank k . Here F [ z ] is the p olynomial ring ov er F a nd F [ z ] n = { ( v 1 , v 2 , . . . , v n ) : v i ∈ F [ z ] } . Suppo se V is a submo dule of F [ z ] n and that { v 1 , . . . , v r } ⊂ F [ z ] n forms a generating set for V . Then V = Imag e M = { uM : u ∈ F [ z ] r } where M =    v 1 . . . v r    ∈ F [ z ] r × n . This M is calle d a ge ner ating matrix of V . A generating matrix G ∈ F [ z ] r × n having rank r is called a gener ator or enc o der matrix of C . A matrix H ∈ F [ z ] n × ( n − k ) satisfying C = ker H = { v ∈ F [ z ] n : v H = 0 } is said to be a c ontr ol matrix of the co de C . 2 Con vo lutional co des from units Let R b e a ring which is a subring o f the ring of matrices F n × n . In particular the gro up r ing F G is a subring of F n × n , where n = | G | , by a n explicit em b edding giv en in [4]. Ther e is no r e striction on F in general but it is as sumed to b e a field her e ; howev er many of the results will hold mo r e generally . Units and zer o-divisors in any ring are defined in the usual manner . Construct R -con volutional codes as follows: 2.1 P olynomial case F or clarity the p olynomial case is consider ed initially although this is a sp ecial case o f the more gener al construction. Suppo se f ( z ) g ( z ) = 1 in R [ z ]. Essentially then the enco der matrix is obtaine d from f ( z ) and the deco der o r control matrix is o btained from g ( z ) using a v ariation on the metho d fo r constr ucting unit- derived code s as fo rm ulated in [3] for non-singula r matrices. Now f ( z ) = ( f i,j ( z )) is a n n × n matrix with en tries f i,j ( z ) ∈ F [ z ]. Similarly g ( z ) = ( g i,j ( z )) is a n n × n matr ix ov er F [ z ]. Suppos e r [ z ] ∈ F [ z ] r and co nsider r [ z ] as an element o f F [ z ] n (b y a dding zeros 2 to the end of it). Then define a mapping γ : F [ z ] r → F [ z ] n by γ : r ( z ) 7→ r ( z ) f ( z ). The code C is the image of γ . Since r [ z ] has zero s in its las t ( n − r ) en tries as a member of F [ z ] n , this means that t he gener ator matrix is the first r r ows of f ( z ) which is an r × n matrix over F [ z ]. Since f ( z ) is inv ertible, this gener ator matrix has ra nk r and is thus the enco der matrix w hich we denote by G ( z ). F or this po lynomial case, G ( z ) is a basic g e nerator matrix – s ee A.1 Theore m in [8]. w ( z ) ∈ F [ z ] n is a co deword if and o nly if w ( z ) g ( z ) is in F [ z ] r , that is , if a nd only if the final ( n − r ) ent rie s of w ( z ) g ( z ) are all 0. Supp ose w ( z ) = ( α 1 ( z ) , α 2 ( z ) , . . . , α n ( z )). Then this co ndit ion is tha t ( α 1 ( z ) , α 2 ( z ) , . . . , α n ( z )) ∗      g 1 ,r +1 ( z ) g 1 ,r +2 ( z ) . . . g 1 ,n ( z ) g 2 ,r +1 ( z ) g 2 ,r +2 ( z ) . . . g 2 ,n ( z ) . . . . . . . . . . . . g n,r +1 ( z ) g n,r +2 ( z ) . . . g n,n ( z )      = 0 The chec k or c ontr ol m a trix H ( z ) of the co de is thus:      g 1 ,r +1 ( z ) g 1 ,r +2 ( z ) . . . g 1 ,n ( z ) g 2 ,r +1 ( z ) g 2 ,r +2 ( z ) . . . g 2 ,n ( z ) . . . . . . . . . . . . g n,r +1 ( z ) g n,r +2 ( z ) . . . g n,n ( z )      This has size n × ( n − r ) and is the ma tr ix consisting of the la st ( n − r ) co lumns of g ( z ) or in other words the matrix o bta ined b y dele tin g the first r columns of g ( z ). Since f ( z ) , g ( z ) ar e units, it is automatic that rank G ( z ) = r and rank H ( z ) = ( n − r ). 2.1.1 Restatement of p olynomial ca se Suppo se then f ( z ) g ( z ) = 1 in R [ z ]. The set- up may be restated as follows: f ( z ) =  f 1 ( z ) f 2 ( z )  g ( z ) =  g 1 ( z ) , g 2 ( z )  where f 1 ( z ) is an r × n ma trix, f 2 ( z ) is an ( n − r ) × n matrix, g 1 ( z ) is an n × r matrix and g 2 ( z ) is an n × ( n − r ) ma tr ix. Then f ( z ) g ( z ) = 1 implies  f 1 ( z ) f 2 ( z )  ×  g 1 ( z ) , g 2 ( z )  = 1 Thu s  f 1 ( z ) g 1 ( z ) f 1 ( z ) g 2 ( z ) f 2 ( z ) g 1 ( z ) f 2 ( z ) g 2 ( z )  = 1 F ro m this it follows that f 1 ( z ) g 1 ( z ) = I r × r , f 1 ( z ) g 2 ( z ) = 0 r × ( n − r ) , f 2 ( z ) g 1 ( z ) = 0 ( n − r ) × r , f 2 ( z ) g 2 ( z ) = I ( n − r ) × ( n − r ) . Thu s f 1 ( z ) is taken as the generator or enco der matrix and g 2 ( z ) is then the chec k or con trol ma trix. Note that bo th f 1 ( z ) , f 2 ( z ) hav e right finite s upport inv erses and thus by Theorem 6 .3 o f [8] the gener ator matrix f 1 is noncatas trophic. Given f ( z ) g ( z ) = 1 b y the gener al descr ibed metho d of unit-derived code in [3] a conv olutio na l co de can b e constr uc ted using any rows of f ( z ). If rows { j 1 , j 2 , . . . , j r } are chosen from f ( z ) then we get an enco ding F r [ z ] → F n [ z ] with generator matrix consisting of these r rows of f ( z ) and chec k/co n trol matrix is obtained by deleting the { j 1 , j 2 , . . . , j r } columns of g ( z ). 3 Cases with f ( z ) g ( z ) = 1 , f ( z ) , g ( z ) ∈ R [ z , z − 1 ], will also in a similar manner pr oduce co nvolutional co des. The next section, Section 2.2, descr ibes the similar pro cess for these in detail. 2.2 More generally Let f ( z , z − 1 ) , g ( z , z − 1 ) ∈ R [ z , z − 1 ] b e such that f ( z , z − 1 ) g ( z , z − 1 ) = 1. Suppo se no w f ( z , z − 1 ) =  f 1 ( z , z − 1 ) f 2 ( z , z − 1 )  g ( z , z − 1 ) =  g 1 ( z , z − 1 ) , g 2 ( z , z − 1 )  where f 1 ( z , z − 1 ) is an r × n matrix , f 2 ( z , z − 1 ) is an ( n − r ) × n matrix, g 1 ( z , z − 1 ) is an n × r matrix and g 2 ( z , z − 1 ) is an n × ( n − r ) matrix. Then  f 1 ( z , z − 1 ) f 2 ( z , z − 1 )  ×  g 1 ( z , z − 1 ) , g 2 ( z , z − 1 )  = 1 Thu s  f 1 g 1 f 1 g 2 f 2 g 1 f 2 g 2  = 1 F ro m this it follows that f 1 ( z , z − 1 ) g 1 ( z , z − 1 ) = I r × r , f 1 ( z , z − 1 ) g 2 ( z , z − 1 ) = 0 r × ( n − r ) , f 2 ( z , z − 1 ) g 1 ( z , z − 1 ) = 0 ( n − r ) × r , f 2 ( z , z − 1 ) g 2 ( z , z − 1 ) = I ( n − r ) × ( n − r ) . Thu s f 1 ( z , z − 1 ) is ta ken a s the ge nerator o r enco der matrix and g 2 ( z , z − 1 ) is then the chec k or c o n trol matrix. It is seen in pa rticular that f 1 ( z , z − 1 ) , f 2 ( z , z − 1 ) hav e r igh t finite s upport inv erses and thus by Theorem 6 .6 of [8 ] the g enerator matrix f 1 is no ncatastrophic. Given f ( z , z − 1 ) g ( z , z − 1 ) = 1 by the general describ ed metho d o f unit-derived co de o f [3] code s any rows of f ( z , z − 1 ) ca n be used to construct a conv olutional. If rows { j 1 , j 2 , . . . , j r } are chosen fro m f ( z , z − 1 ) then an enco ding F r [ z ] → F n [ z ] is obta ine d with genera tor matrix consisting of these r r o ws of f ( z ) and chec k/control matrix obtained by deleting the { j 1 , j 2 , . . . , j r } columns of g ( z ). 2.2.1 P articular case Suppo se f ( z ) g ( z ) = z t in R [ z ]. Then f ( z )( g ( z ) /z t ) = 1. Now ( g ( z ) /z t ) in volv es negative p ow ers o f z but has finit e support. The encoder matrix is obtained fro m f ( z ) and th e decoder or control matrix is obtained from ( g ( z ) /z t ) using the method as form ulated in Section 2.2. It is also possible to consider ( f ( z ) / z i )( g ( z ) /z j ) = 1 with i + j = t and to derive the gene r ator matrix from ( f ( z ) /z i ) and the chec k/control matrix from ( g ( z ) /z j ). The control matrix co n tains negative p o wers of z but a p olynomial control matrix is ea sy to obtain from this. Note that z − n are units worth co nsidering in R [ z , z − 1 ] but that other e le men ts in R [ z , z − 1 ] may hav e inv er s es with infinit e s upport and the inv erses are thus outside R [ z , z − 1 ]. How ever in some cases m X i = − t α i z i ∈ R [ z , z − 1 ] has a n in verse in R [ z , z − 1 ], for example in certain ca ses when the α i are nilp oten t, and her e also conv olutiona l co des may b e defined with (direct) noncatas tr ophic g enerator matrices. All these a re cases of f ( z , z − 1 ) × g ( z , z − 1 ] = 1 ∈ R [ z , z − 1 ] but may be worth considering o r iginally from po lynomials for the constructio n. 2.2.2 Unin teresting zero-divisors In [3] units a nd zer o -divisors in group rings are used to construct co des. Zero- divisors in R [ z ] are not to o interesting: Supp ose uw = 0 in R [ z ] a nd u is an element of leas t degree so that uw = 0. Then w or 4 u has degr e e zero; if w has degree 0 then it is a zer o-divisor of ea c h co efficient of u and if u has degree zero then it is a z ero-divisor of each co efficien t of w . Thu s if w e require zer o-divisor co des in R [ z ] w e are looking at direct sums of zero-divisor codes in R . Using units in R [ z ] to construct co des is far more pr oductive. 2.3 Group ring matrices In the constructio ns of Section 2.1 or in the mor e general Se c tio n 2.2, R is a subring of F n × n . Supp ose now R = F G is the g roup ring of the gr o up G ov er F . The group r ing R G is a subring o f F n × n using an explicit corr e s pondence b e t ween the group ring R G and the ring of R G - ma trices, see e. g. [4]. Thu s the metho ds of Section 2.1 and/or Section 2.2 may be used to define convolutional co des using group rings R = F G as a s ubr ing of F n × n and then forming R [ z , z − 1 ] ∼ = RC ∞ , which is the group ring ov er C ∞ with co efficient s from the gr oup r ing R = F G . T o obtain units in R [ z , z − 1 ] (which includes R [ z ]) we a re lead to consider zero -divisors and units in R = F G . R = F G is a r ic h s ource of zero-div isors, and units, and conse quen tly R [ z , z − 1 ] is a rich source of units. There a re methods av a ilable for constructing units and zero- divisors in F G . If F is a field, every no n-zero element of F G is either a unit or a zero-divisor. What is req uired ar e units in R [ z ], where R = F G , a gr oup ring , and these can b e obtained by the use of zero-divis ors and units in R as c o efficients of the p o wers of z . In what follo ws be a r in mind that in R [ z , z − 1 ] it is poss ible and desirable that R has zero-div isors and units, as when R is a gro up ring. 3 Con vo lution co des from group rings Suppo se then n X i = − m α i z i × n X j = − m β j z j = 1 in the gr oup ring RC ∞ = R [ z , z − 1 ] with α i ∈ R and C ∞ generated by z . By multiplying through by a power of z this is then n X i =0 α i z i × n X j = − m β j z j = 1. The case with m = 0 gives p olynomials o ver z . Here we have n X i =0 α i z i × t X i =0 β j z j = 1 wher e α n 6 = 0 , β t 6 = 0 and lo oking at the co efficien t of z 0 it is clear that we must also hav e α 0 6 = 0 , β 0 6 = 0. This can be considered an an equa tion in RC ∞ with non-negative pow ers. So lut ions may be used to construct conv olutional co des. By look ing at the highest and low est coefficients we then ha ve that α 0 × β 0 = 1 and α n × β t = 0 . Thu s in par ticular α 0 is a unit with inverse β 0 and α n , β t are zer o divisors. Solutions of the general equation n X i =0 α i z i × n X j = − m β j z j = 1 can also be used to for m convolutional co des and p olynomial gener ator ma trices may be derived from these. 4 Examples 4.1 A prototype example Let R = Z 2 C 4 . The n α 0 = a + a 2 + a 3 satisfies α 2 0 = 1 a nd α 2 = a + a 3 satisfies α 2 2 = 0. Thu s w = α 0 + α 1 z + α 2 z 2 in RC ∞ satisfies w 2 = α 0 α 0 + z ( α 0 α 1 + α 1 α 0 ) + z 2 ( α 0 α 2 + α 2 1 + α 2 α 0 ) + z 3 ( α 1 α 2 + α 2 α 1 ) + z 4 ( α 2 α 2 ) = 1 + z 2 α 2 1 , s ince the α i commute. Now require tha t α 2 1 = 0 and then w 2 = 1. 5 In particular letting α 1 = α 2 implies that w 2 = 1 . Howev er, just to be different , consider α 1 = 1 + a 2 and then also α 2 1 = 0 . Now α 0 corres p onds to the matrix     0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0     , α 2 corres p onds to the matrix     0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0     and α 1 corres p onds to the matrix     1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1     . T ake the first tw o rows of w to gener a te a conv olutiona l co de and then the la st t wo columns of w is the control matrix o f this co de. This gives the following genera tor matrix : G =  0 1 1 1 1 0 1 1  +  1 0 1 0 0 1 0 1  z +  0 1 0 1 1 0 1 0  z 2 The control matrix is :     1 1 1 1 0 1 1 0     +     1 0 0 1 1 0 0 1     z +     0 1 1 0 0 1 1 0     z 2 The co de has le ngth 4 and dimensio n 2. It may be shown that the fr ee distance of this co de is 6. This can b e gene r alised. 5 Con vo lutional co des from nil p oten t ele men ts The following tw o theorems are useful in co nstructing new clas ses of conv olutional co des. Theorem 5.1 L et R = F G b e the gr oup ring of a gr oup G over a field F with char acteristic 2 . Supp ose α i ∈ R c ommute. L et w = n X i =0 α i z i ∈ RC ∞ . Then w 2 = 1 if and only if α 2 0 = 1 , α 2 i = 0 , i > 0 . Pro of: The pr oof of this is stra igh t-forward and is o mitted.  The following is a generalisa tion o f Theorem 5.1; its pro of is a ls o straight-forw ar d a nd is omitted. Theorem 5.2 L et R = F G b e the gr oup ring of a gr oup G over a field F with char acteristic 2 . Supp ose α i ∈ R c ommute. L et w = n X i =0 α i z i ∈ RC ∞ . Then w 2 = z 2 t if and only if α 2 i = 0 , i 6 = t and α 2 t = 1 . T o then construct conv olutiona l co des proceed as follows. Find elements α i with α 2 i = 0 and units u with u 2 = 1 in the group ring R . Then form units in R [ z ] or R [ z , z − 1 ] using Theorem 5 .1 or Theo- rem 5 .2 . F rom these units, convolutional co des are defined using the methods describ ed in Section 2 .1 or Section 2.2. 5.1 Examples 1 Consider now α 0 = a + a 2 + a 3 and for i > 0 define α i = a + a 3 or α i = 0 in the gr oup ring R = Z 2 C 4 . Then α 2 0 = 1 and α 2 i = 0 , i > 0. W e could also take α i = 1 + a 2 . Define w ( z ) = n X i =0 α i z i in R C ∞ . B y Theorem 5.1, w 2 = 1. 6 The matrix corresp onding to α 0 is     0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0     and the matrix co rresponding to α i , i 6 = 0 is     0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0     or else is the zero matrix. Now spe c if y tha t the first tw o rows of w give the generato r matrix and from this it follo ws that the last tw o co lumns of w is a co n trol matrix. This gives the following genera tor matrix : G =  0 1 1 1 1 0 1 1  + δ 1  0 1 0 1 1 0 1 0  z + δ 2  0 1 0 1 1 0 1 0  z 2 + . . . + δ n  0 1 0 1 1 0 1 0  z n where δ i = 1 when α i 6 = 0 and δ i = 0 when α i = 0. The control matrix is : H =     1 1 1 1 0 1 1 0     + δ 1     0 1 1 0 0 1 1 0     z + δ 2     0 1 1 0 0 1 1 0     z 2 + . . . + δ n     0 1 1 0 0 1 1 0     z n . The co de ha s length 4 and dimension 2 . The free distance is at lea st 6 for any n ≥ 2 and in many cases it will be larger. Polynomials used for genera ting cyclic linear codes suita bly conv erted to polynomials in R [ z ] prov e particularly useful and amena ble – see fo r example Section 7 b elo w. 5.1.1 P articular Example The (4 , 2) conv olutiona l co de with generator and chec k matr ic es a s follo ws has free distance 8. G =  0 1 1 1 1 0 1 1  +  0 1 0 1 1 0 1 0  z +  0 1 0 1 1 0 1 0  z 3 +  0 1 0 1 1 0 1 0  z 4 H =     1 1 1 1 0 1 1 0     +     0 1 1 0 0 1 1 0     z +     0 1 1 0 0 1 1 0     z 3 +     0 1 1 0 0 1 1 0     z 4 6 Direct pro ducts: T urb o-effect Examples o f conv olutional co des formed using α i with α 2 i = 0 in F G hav e b een pr oduced. Co nsider now F ( G × H ) and let w = β × α i for a ny β ∈ F H . Then w 2 = β 2 α 2 i = 0. This expa nds eno rmously the ra nge of av a ilable elemen ts who s e square is zer o. Note also that ov er a field of characteris tic 2 if α 2 = 0 = γ 2 then ( α + γ ) 2 = 0 . F or example in Z 2 C 2 the element 1 + a was used where C 2 generated by a . Then in Z 2 ( G × C 2 ) consider α = β (1 + a ) for any β ∈ Z 2 G . Then α 2 = 0. A s imple example of this is Z 2 ( C 2 × C 2 ) w he r e α = (1 + a ) b + (1 + b ) a = a + b . The matrix of a + b is  A B B A  where A =  0 1 1 0  and B =  1 0 0 1  . In for ming (4 , 2) conv olutional co des we would only use the to p half of the matrice s , i.e. P =  0 1 1 0 1 0 0 1  . Note that in this enco ding the vector ( γ , δ ) is mapp ed to ( γ , δ ) P =  δ γ γ δ  . This is like an int er weaving of t wo co des. 7 T o get a per mutation effect, use the direct product with S n , the permutation or symmetric g roup on n letter s . 7 (2,1) co des See [8] for ex amples of (2 , 1) optimal co des up to deg ree 10. Thes e can b e repro duced algebr aically a nd prop erties derived using the metho ds developed here . F urther new (2 , 1) conv olutional co des and ser ie s of conv olutional (2 , 1) ar e cons tr ucted in this section as an applicatio n o f the general metho ds descr ibed a b ov e. The free distances ca n often b e determined algebraic ally and co des to a prescib ed free distanc e ca n be constructed by using Theo r em 7.3 b elo w. Let F b e a field of characteristic 2 a nd R = F C 2 , where C 2 is generated b y a . Consider elemen ts α i ∈ R , i > 0, wher e either α i = 1 + a o r α i = 0. Then α 2 i = 0 . Let α 0 = 1 in R and define w = α 1 + α 0 z + α 2 z 2 + . . . + α n z n . Then w 2 = z 2 and hence w × ( w/z 2 ) = 1. Thu s w can be used to define a (2 , 1) conv olutiona l co de. More ge ne r ally let t b e an in teger, 0 ≤ t ≤ n , and define w = n X i =0 β i z i where β i = α i , i 6 = t , β t = 1. Then w 2 = z 2 t gives that w × ( w /z 2 t ) = 1 . Th us w can be used to define a convolutional (2 , 1) co de. The ca se α 0 = β 1 is a sp ecial case. Now deter min e the co de by choosing the first r o w of the matrix of w to b e the generator /encoder matrix and then the la st column of w /z 2 t is the control matrix. The matrix of α i is  1 1 1 1  when α i = 1 + a and is the ze ro 2 × 2 matrix when α i = 0 . . Define δ i = 1 when α i 6 = 0 and i 6 = t ; δ i = 0 when α i = 0 a nd i 6 = t ; and define δ t (1 , 1 ) to be (1 , 0). Then the enco der matrix of the co de is G = (1 , 1) + δ 1 (1 , 1 ) z + δ 2 (1 , 1 ) z 2 + . . . + δ n (1 , 1 ) z n and with H =  1 1  + δ 1  1 0  z + δ 2  1 1  z 2 + . . . + δ n  1 1  z n , the control matrix is H /z 2 t . The gene r ator matrix G obtained in this w ay is noncatastrophic a s it has a right finite w eight inv erse – see Theorems 6 .3 and 6.6 in [8]. F or n = 2 w e get as an example the code with th e generator matrix G = (1 , 1) + (1 , 0) z + (1 , 1) z 2 . This co de has free distance 5 which is optimal. It is precisely the (2 , 1 , 2 , 5) c ode a s descr ibed in [8], pa g e 1085. Theorem 7.1 G has fr e e distanc e 5 . Pro of: Consider t X i =0 β i z i G , with β i ∈ Z 2 and β t 6 = 0. In determining free distance we may consider β 0 6 = 0. The co efficien ts of z 0 = 1 and z t +2 are (1 , 1 ), and also (1 , 0 ) o ccurs in the expression for at least one other co efficien t. Thus the free distance is 2 + 2 + 1 which is a tt ained by G .  The ab ov e pr o of illus trates a general metho d for proving free distance or getting a low er b ound on the free dis ta nce. F or example wher ev er (1 , 0) app ears in a s um making up a coefficient it will contribute a distance of a t least 1 a s the other no n-zero coe fficie nts, all (1 , 1), will add up to (1 , 1) or (0 , 0). The chec k matrix for this co de is 0 @ 1 1 1 A + 0 @ 0 1 1 A z + 0 @ 1 1 1 A z 2 z 2 =  1 1  +  0 1  z − 1 +  1 1  z − 2 . F or n ≥ 3 it ma y be verified directly by simila r alge br aic metho ds tha t the free distance is at least 6. Appropriate choices of the α i will give bigger free distances. See Theorem 7.3 b elow. F or n = 3 , andδ 2 = 1 = δ 3 a (2 , 1 , 3 , 6) co n volutional co de is o bta ined which is also optima l. Thus a degree 3 optimal distance 6 is given by the enco der matrix G = (1 , 1) + (1 , 0) z + (1 , 1) z 2 + (1 , 1) z 3 and the con tro l matrix is H /z 2 =  1 1  /z 2 +  1 0  /z +  1 1  +  1 1  z . It is clear that H is also a control matrix. 8 The next case is (2 , 1 , 4 ) o f degree 4. The optima l distance of one of these is 7 . Consider w = α 1 + α 0 z + α 1 z 3 + α 1 z 4 , wher e α 1 = 1 + a and α 0 = 1 in Z 2 C 2 . Then w 2 = z 2 and thus w gives the enco der matrix and w/ z 2 gives the check matr ix. The enco der matrix is G = (1 , 1) + (1 , 0) z + (1 , 1) z 3 + (1 , 1) z 4 . Call this co de C . Theorem 7.2 The fr e e distanc e of C is 7 . Pro of: Co nsider ( t X i =0 β i z i ) G , with β i ∈ Z 2 . In determining free dista nc e w e may co nsider β 0 6 = 0 and β t 6 = 0. The coefficients o f z 0 (= 1) and z t +4 are b oth (1 , 1). If there are more th an t wo no n- zero β i in the sum then (1 , 0) o ccurs in a t least three coefficients g iving a distance of 2 + 2 + 3 = 7 a t least. It is now necessary to consider the case when there ar e just tw o β i in the sum. It is ea sy to see then that at least three of the co efficien ts of z i are (1 , 1), and (1 , 0) or (0 , 1 ) is a coe fficien t of another. T hus the free distance is 7.  Consider the next few degr e es. Let α = 1 + a, α 0 = 1 in F C 2 where F has characteristic 2. 1. deg 5 : w = α + α 0 z + αz 3 + αz 4 + αz 5 ; gives a free distance of 8 . 2. deg 6 : w = α + αz 2 + αz 3 + α 0 z 4 + αz 5 + αz 6 . This gives a free distance of 10. 3. Consider for example the following degr ee 1 2 elemen t. w = α + αz 2 + αz 4 + αz 5 + αz 6 + α 0 z 9 + αz 10 + αz 11 + αz 12 Note that this resem bles the p olynomial used for the Golay (23 , 12) co de – see e.g. [1] page 119 . The difference is that a z 12 has b een added and the coefficient o f z 9 app ears with co efficient α 0 and not 0 as in the Golay co de. It is p ossible to play around with this by placing α 0 as the co efficient of other powers of z in w . W e th us study the best p erformance of co n volutional co des derived from w = t X i =0 α i z i where some α t = 1 ∈ F C 2 , and all the other α i are either 0 or else 1 + a in F C 2 . T ry to choose the α i as one would for a linear cy c lic code so as to maximise the (free) distance. The set-up indicates we should lo ok at exis ting cyclic co des and form conv olutional co des by mimic king the gener ating polynomia ls for the cyc lic co des. 7.1 F rom c yc lic c o des to conv olutional co des Suppo se now C is a (linear) cyclic ( n, k , d 1 ) co de ov er the field F o f c haracteris tic 2. Supp ose a lso that the dual of C , denoted ˆ C , is an ( n, n − k , d 2 ) co de. Let d = min ( d 1 , d 2 ). Suppo se f ( g ) = r X i =0 β i g i , with β i ∈ F, ( β r 6 = 0), is a generating p olynomial for C . In f ( g ), as sume β 0 6 = 0. Consider f ( z ) = r X i =1 α i z i where now α i = β i α with α = 1 + a in F C 2 or else α i = 0. Replace some α i , say α t , by 1 or a (considered a s mem b ers of F C 2 ). So assume f ( z ) = r X i =0 α i z i with this α t = 1 and other α i = β i α so that α i = 1 + a or α i = 0 (for i 6 = t ). It is also a llo wed to let α t = a . Then f ( z ) 2 = z 2 t and th us f ( z ) × ( f ( z ) /z 2 t ) = 1. W e now use f ( z ) to generate a conv olutional co de by taking just th e fir st rows o f the α i . Thus the ge ner ating matrix is ˆ f = r X i =0 ˆ α i z i where ˆ α i is the fir st row of α i . 9 Lemma 7.1 L et G b e a gener ator matrix of a line ar c o de C and su p p ose t h e du al c o de of C , ˆ C , has distanc e d . Then n o ro w of G is a c ombination o f less than d − 1 other r ows of G . Pro of: Now G T is the chec k matrix of ˆ C . Since ˆ C ha s distance d any d − 1 c o lumns of G T are linearly independent – see e.g. [1], Coro lla ry 3.2.3, page 52 . T hus no column of G T is a combination of les s than d − 1 o ther columns of G T . Hence no row of G is a co m bina tion of less tha n d − 1 other r o ws of G .  Lemma 7.2 L et w = n X i =1 α i (1 , 1 ) + α (1 , 0) with α 6 = 0 . Then at le ast one c omp onent of w is not zer o. Pro of: Now w = ( n X i =1 α i + α, n X i =1 α i ). Since α 6 = 0 it is clear that one co mponent of w is not zer o.  A similar r esult holds for w = n X i =1 α i (1 , 1 ) + α (0 , 1). F or the following theore m assume the inv ertible element α 0 do es not o ccur in the fir s t or the last po sition of f ; if it do es occur in o ne of these p ositions, a s imilar result holds but the free distance is po ssibly less by 1. Theorem 7.3 L et C denote t he c onvolutional c o de with gener ator matrix ˆ f . Then the fr e e distanc e of C is at le ast d + 2 . Pro of: Consider w = t X i =0 β i z i ˆ f and we wish to show that its free distance is ≥ d + 2 . In calculating the free distance of w we can as s ume β 0 6 = 0 and w e also naturally assume β t 6 = 0. Let f d ( w ) denote the fre e distance of w . Let w 1 = t X i =0 β i z i . The s upport of w 1 , su pp ( w 1 ), is the num b er of non-z e r o β i . Suppo se then supp ( w 1 ) ≥ d . Then in w , α 0 app ears with the co efficien t of z i , for at least d different i with 0 < i < t + r . Also the co efficient of 1 = z 0 is β 0 (1 , 1 ) and the co efficien t of z t + r is β t (1 , 1 ) and each of these have distance 2. Then by Lemma 7.2, w has free distance at least d + 2. Consider f ( g ) = r X i =0 β i g i and H ( g ) = f ( g )( l X i =0 δ i g i ), with l ≤ k − 1 where k is the ra nk of the cyclic co de. Then as this cyc lic co de has distance d 1 , H ( g ) = n − 1 X i =0 γ i g i has supp ort at least d 1 . Now H ( z ) = f ( z )( l X i =0 δ i z i ) is such that the sum of the co efficien ts of z i , z i + n , . . . is γ i for each i . Hence if γ i 6 = 0, a t least one of th e co efficien ts of z i , z i + n , . . . is not 0. Since H ( g ) has supp ort d 1 , this implies that H ( z ) has s upp ort a t least d 1 . Hence w has free dista nce at lea st ( d 1 − 2) + 2 × 2 = d 1 + 2 ≥ d + 2 when t ≤ ( k − 1). Assume then in w that t ≥ k . and that supp ( w 1 ) < d . If supp ( w 1 ) = 1 then clea rly f d ( w ) ≥ ( r − 2) + 4 = r + 2 ≥ d + 2. Assume by induction that a sum s uch as w of less than t elements with supp ort less than d has free distance at least d + 2. Consider f ( g ) = r X i =0 β i g i and H ( g ) = f ( g )( l X i =0 δ i g i ), where t > k − 1. Now as C has rank k , f ( g ) g k = k − 1 X i =0 δ i f ( g ) g i . Thus multiplying through b y g t − k implies f ( g ) g t = 10 k − 1 X i =0 δ i g i + t − k f ( g ) = t − 1 X j = t − k δ j − ( t − k ) f ( g ) g j . Now a s ˆ C has distance d 2 the supp ort of k − 1 X i =1 δ i f ( g ) g i and hence of t − 1 X j = t − k δ j − ( t − k ) f ( g ) g j is at leas t d 2 − 1 by Lemma 7 .1. Now t − 1 X i =0 β i z i ˆ f has supp ort at mos t d − 2 as w ha s suppor t at most d − 1. Then w = t X i =0 β i z i ˆ f = t − 1 X i =0 β i z i ˆ f + β t z t ˆ f = t − 1 X i =0 β i z i ˆ f + β t t − 1 X j = t − k δ j − ( t − k ) ˆ f z j = t − 1 X i =0 ω i ˆ f z i and this sum is of non-zero s upport. Thus b y induction the f d ( w ) ≥ d + 2.  The free distance may b e big ger than d + 2 ; a n upper bound is 2 d − 1. The free distance also dep ends on wher e the in vertible α 0 is pla c e d in the expre ssion for f . Pla ced near the ‘centre’ will possibly giv e the b est free dis tance. It is worth noting tha t if the supp ort o f the input elemen t is ≥ t then the free dista nc e is a t least t + 2; this may b e seen from the pro o f of Theor em 7.3. Thus it is p ossible to av oid shor t distance co dewords by ensur ing tha t the input elements ha ve sufficien t suppor t – this could be do ne by , for example, taking the complement of a n y element with small supp ort. The b est choice for C is probably a self-dual co de as in this ca se d 1 = d 2 = d . There exist self-dual co des o f a rbitrary large distances. See als o [5] for many cons tructions of self-dual co des. These c o n volutional co des can be co nsidered to b e self-dual type convolutional co des in the sense that f ( z ) determines the g enerator matrix and f ( z ) /z 2 t determines the co ntrol matrix. 8 (2m,1) co des The previous section Section 7 ca n b e g eneralised to pro duce con volutional co des of smaller ra te (2 m, 1) but with muc h bigger fr e e distance. Essentially the free distance is m ultiplied by m ov er that o btained for similar (2 , 1) co des. The gr oup to consider is C 2 m generated by a . Assume m is o dd although similar results may b e obtained when m is e ven. Let α = 1 + a + a 2 + . . . + a 2 m − 1 and α 0 = 1 + a 2 + . . . + a 2 m − 2 . Then α 2 = 0 and α 2 0 = 1 a s α 0 has o dd supp ort. Define as b efore f ( z ) = r X i =1 α i z i where now α i = β i α in Z 2 C 2 m or else α i = 0. Replace some α i , say α t , by α 0 . Then f ( z ) 2 = z 2 t and f ( z )( f ( z ) /z 2 t ) = 1 . Thus use f ( z ) to define a conv olutiona l co de C by taking the firs t row of the α i . F or ex ample G ( z ) = (1 , 1 , 1 , 1 , 1 , 1) + (1 , 0 , 1 , 0 , 1 , 0) z + (1 , 1 , 1 , 1 , 1 , 1) z 2 defines a (6 , 1) conv olu- tional co de which has fr ee distance 15. G ( z ) = (1 , 1 , 1 , 1 , 1 , 1 ) + (1 , 0 , 1 , 0 , 1 , 0) z + (1 , 1 , 1 , 1 , 1 , 1) z 3 + (1 , 1 , 1 , 1 , 1 , 1) z 4 defines a conv olutiona l co de which has free distance 21 . A theorem similar to Theorem 7.3 is also true:Let f ( g ) denote the generato r matrix of a cyclic co de with distance d 1 and whose dua l co de ha s distance d 2 . Let d = min( d 1 , d 2 ) and let C deno te the conv olutional code obtained from f ( z ) where the co efficien ts of f ( g ) hav e been r eplaced b y α i in all but one co efficient which has been replaced b y α 0 and the first row o f each co efficien t is used. Assume in the following theorem that α 0 is not in first or last co efficient . Theorem 8.1 The fr e e distanc e of C is at le ast md + 2 m . 11 9 Higher rates The metho ds of Section 7 can a lso be gener alised to pro duce higher rate conv olutional co des. Consider achieving a r ate of 3 / 4. In C 4 generated by a , define α = 1 + a and α 0 = 1 . Then α 4 = 0 and α can be used to define a co de of r ate 3 / 4 and distance 2 . Now α has ma trix     1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1     and the first three rows of this A =   1 1 0 0 0 1 1 0 0 0 1 1   generates a (4 , 3 , 2) co de. Now the matrix of α 0 is I 4 × 4 , the iden tity 4 × 4 matrix and let B denote the first three ro ws of I 4 × 4 . Lemma 9.1 L et x 6 = 0 b e a 1 × 3 ve ctor. Then x ( A + B ) is not the zer o ve ctor and thus x ( A + B ) has distanc e at le ast 1 . Pro of: Now ( α + 1) 4 = α 4 + 1 = 1 and so ( α + 1) is a non-singular matrix . Th us in particula r the first three rows of the matrix of ( α + 1) ar e linearly indep enden t. The firs t three ro ws of α + 1 precis ely constitutes the matrix A + B . Thus x ( A + B ) is no t the zer o v ector . Another wa y to lo ok at this is that α + 1 = a but it is useful to lo ok at the more genera l way in Lemma 9.1 for further developments.  Corollary 9.1 If x A + y B = 0 then x 6 = y . F or m con volutional (4 , 3) co des as follows. Let f ( z ) = n X i =0 α i z i where α i = α or α i = 0 except for α t = 1 for s ome t, 1 < t ≤ n . W e could also use α 1 = α t = 1 but this generally g iv es smaller dista nce codes. Then f ( z ) 4 = z 4 t and so f ( z ) × ( f ( z ) 3 /z 4 t ) = 1. Thus use f ( z ) to generate the code and ( f ( z ) 3 /z 4 t ) to chec k/control the co de. T ake the firs t three rows of the ma trix of f ( z ) to gener ate a (4 , 3) co de and delete the last thr ee columns ( f ( z ) 3 /z 4 t ) to form the control matrix. Thu s G ( z ) = n X i =0 ˆ α i z i is the generator matrix wher e ˆ α i is the first three rows of the matrix of α i . In Section 7 we had the situation that when α 0 o ccurred in a ny co efficien t then it co n tributed a distance of 1, so tha t when the supp ort of G is s then α 0 will contribute a free distance of s . Here w e us the fact that if α 0 o ccurs then it will contribute a dis tance of at leas t 1 unless its co efficient equals the sum of the co efficients in the o th er non-zero α i which occur with it in the sa me co efficien t of z j . 9.1 Examples The g e nerator matrix G =   1 1 0 0 0 1 1 0 0 0 1 1   +   1 0 0 0 0 1 0 0 0 0 1 0   z +   1 1 0 0 0 1 1 0 0 0 1 1   z 2 defines a (4 , 3) co n volutional co de. It may b e shown that its fre e distance is 5. The pro of is similar to the pro of of Theor em 7.1 but a ls o us ing Lemma 9.1. 12 The chec k matrix for the co de is easy to write out. Consider n = 3, and G =   1 1 0 0 0 1 1 0 0 0 1 1   +   1 0 0 0 0 1 0 0 0 0 1 0   z +   1 1 0 0 0 1 1 0 0 0 1 1   z 2 +   1 1 0 0 0 1 1 0 0 0 1 1   z 3 This is a (4 , 3) conv olutiona l co de and its free distance is 6. The next example is G =   1 1 0 0 0 1 1 0 0 0 1 1   +   1 0 0 0 0 1 0 0 0 0 1 0   z +   1 1 0 0 0 1 1 0 0 0 1 1   z 3 +   1 1 0 0 0 1 1 0 0 0 1 1   z 4 This has fr ee distance 7. This may be proved similar to Theo rem 7.2 using L e mm a 9.1. It is then p ossible to proceed as in Section 7 to inv estigate further degrees (memories) with r ate 3 / 4 . 9.2 P olynomial In cas e s where a polyno mia l generator and polynomial righ t inv erse for this g enerator a re require d, insist that α 0 = 1. This g iv es slightly less free distance but is interesting in itself. F or exa mp le consider the enco der matr ix G = (1 , 0) + δ 1 (1 , 1 ) z + . . . + δ n (1 , 1 ) z n and the control matrix is H =  1 0  + δ 1  1 1  z + . . . + δ n  1 1  z n . Her e δ i = 0 or δ i = 1 . This co de has free distance 4 for n = 2. F or n ≥ 2 the free distance w ill dep end on the choice of the δ i . As already no ted, the c hoices wher e the z -po lynomial co rrespo nds to a known cyclic code polyno mia l deserves particular attention. W e may also increas e the size o f the field as for example as follows. Consider now R = GF (4) C 2 , the g roup ring o f the cyc lic g roup of orde r 2 over the field of 4 elements. Define α 0 = ω + ω 2 g , α 1 = ω + ω g , α 2 = ω 2 + ω 2 g , where ω is the primitive element in GF (4) which satisfies ω 2 + ω + 1 = 0 , ω 3 = 1 . Then α 2 0 = ω 2 + ω 4 = ω 2 + ω = 1 and α 2 1 = α 2 2 = 0 . Thus w = α 0 + α 1 z + α 2 z 2 satisfies w 2 = 1 and can be used to define a con volutional co de of length 2 and dimension 1. The enco der matrix is then G = ( ω , ω 2 ) + δ 1 ( ω , ω ) z + δ 2 ( ω 2 , ω 2 ) z 2 + ... + δ n ( ω i , ω i ) z n and the control matrix is H =  ω 2 ω  + δ 1  ω ω  z + . . . + δ n  ω i ω i  z n . The de gr e e of a conv olutional co de with e ncoder matrix G ( z ) is defined to b e the maximal degree of the full k × k size minors of G ( z ) where k is the dimens io n; see [1]. The maximum free distance of a length 2 , dimension one, deg ree δ co de ov er any field is by [11], 2 δ + 2. Consider the case n = 2. The enco der matrix is then G = ( ω, ω 2 ) + ( ω , ω ) z + ( ω 2 , ω 2 ) z 2 . The deg ree of this co de is δ = 2 since the dimensio n is 1. L e t G ′ = (1 , ω ) + (1 , 1) z + ( ω , ω ) z 2 so that ω G ′ = G . Theorem 9.1 The fr e e distanc e of t h is c o de is 6 and so is thus a maximum distanc e sep ar able c onvolu- tional c o de. Pro of: Consider combinations ( α 0 + α 1 z + . . . + α t z t ) G and we wish to show that this ha s (free) distance 6. W e may assume α 0 6 = 0. It is clear when t = 0 that w has a distance o f 6 and so in par ticular a distance of 6 is attained. Since also ω is a factor of G we may now co nsider the minimum distance of w = ( α 0 + α 1 z + . . . + α t z t ) G ′ with α 0 6 = 0 , α t 6 = 0 and t > 0. The co efficien t of z 0 is α 0 (1 , ω ); the co efficien t of z t +2 is α t ( ω , ω ), the co efficient of z t +1 is α t (1 , 1 ) + α t − 1 ( ω , ω ) and the coe fficie n t of z t is α t (1 , ω ) + α t − 1 (1 , 1 ) + α t − 2 ( ω , ω ) when t ≥ 2 and the co efficien t of z is α 1 (1 , ω ) + α 0 (1 , 1 ) and this is also the case when t = 1. Case t ≥ 2: If α t 6 = α t − 1 ω then the co efficien t of z t +1 has distance 2 giving a distance of 6 with 2 coming from each of the co efficien ts of z 0 , z t +1 , z t +2 . If α t = α t − 1 ω the co efficien t of z t is α t − 1 ( ω + 1 , ω 2 + 1 ) + α t − 1 ( ω , ω ); in any case this has distance ≥ 1. Also the co efficien t of z has distance ≥ 1. Thu s the total distance is at least 2 + 1 + 1 + 2 = 6. 13 Case t = 1. If α 0 ω 6 = α 1 then the co efficien t of z 2 has distance 2 and thus g et a distance of 2 + 2 + 2 = 6 for the co efficient s of z 0 , z 2 , z 3 . If α 0 ω = α 1 then the co efficient of z is α 1 (1 , ω )+ α 0 (1 , 1 ) = α 0 ( ω +1 , ω 2 +1) which has distance 2 . Thus a ls o we get a dista nc e of 2 + 2 + 2 = 6 from c o efficients of z 0 , z , z 3 . Note that the pro of dep ends on the fact that { 1 , ω } is linearly indep enden t in GF (4 ).  9.2.1 Bigger fields It will b e necess a ry to work ov er bigger fields to get length 2, dimensio n 1, maximal distance separ able conv olutional co des of higher degree. Consider F = GF (2 n ) with gener ating elemen t ω sa tisfying ω n + ω + 1 = 0. Then w 0 = ω + ω n a in F C 2 , where C 2 is generated b y a satisfies w 2 0 = ω 2 + ω 2 n = 1 since ω n = ω + 1 and w i = ω i + ω i , defined for i > 0 , satisfies w 2 i = 0. A generating element is then formed fro m these w i . Consider w ( z ) = w 0 + δ 1 w i 1 z + . . . + δ n w i n z n where w i j is some w i and δ i ∈ { 0 . 1 } . Then w ( z ) 2 = 1 and is then used to define a conv olutiona l co de of length 2 and dimensio n 1. The w 0 can be taken as the co efficien t of an y z t in the definition of w ( z ) and co n volutional co des ar e similarly defined. The further study of these co des is not included her e. 10 General rank considerations Let w ( z ) = t X i =0 α i z i where α 2 i = 0 , i 6 = t, α 2 t = 1 with the α i in some group ring RG . Suppos e the α i commute and that R has characteristic 2. Then w ( z ) 2 = z 2 t . Consider the ranks of the non-zero α i in deciding whic h rows of w to choos e with which to constr uct the conv olutiona l code. F or exa mp le if the non-zer o α i satisfy rank α i = 1 / 2 | G | = m we choo se the matrix with just ha lf the rows of the matrix o f eac h α i . Many g oo d co des may b e pro duced this way . It is p ossible to have more than o ne α t satisfying α 2 t = 1 in w ( z ) but then the genera tor matrix pro duced can b e catas trophic, although a v alid co de may still b e defined. 10.1 Example Let u = 1 + h ( a + a 2 + a 3 ) in Z 2 ( C 4 × C 2 ). Then u 2 = 0 a nd rank u = 4. Define w = u + z + u z 2 . Then w 2 = z 2 and w is used to define a (8 , 4 ) convolutional co de. The g enerator matrix is G = ( I , B ) + ( I , 0 ) z + ( I , B ) z 2 where B =     0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0     . Now ( I , B ) has distance 4. Any com bination of ( I , B ) , ( I , 0) has distance 1 at least as B is non-singular. Thus consider ( t X i =0 β i z t ) G . The highest and low est pow er of z has distance 4 and there is a power of z in be tween which has dis tance 1 s o altog ether we get a free distance o f 9. The degree of the co de is 8. This can b e extended. It can a lso be ex tended b y finding higher dimensiona l u with u 2 = 0. See Section 1 4 for further development of these ideas . 10.2 Higher rates with nilp oten t elemen ts So far we hav e used α i with α 2 i = 0 a nd this generally g iv e rate 1/2 co n volutional co des. W e no w lo ok at elements α with α 4 = 0 with which to pro duce c o n volutional rate 3/4 co des. See [5] for where such elements are used to pro duce dual-containing co des. See Section 9 for some pr eliminary examples on these. 14 Suppo se then w = n X i =0 α i z i in F G where α 4 i = 0 , i 6 = t a nd α 4 t = 1 , 1 ≤ t ≤ n . Supp ose also F has c hara cteristic 2 and that the α i commute. Then w 4 = z 4 t . Thus w is used to genera te a 3/4 rate conv olutional c o de by tak ing the first 3/ 4 of the rows of the α i ; then w 3 /z 4 t will b e the con trol matr ix using the last 1/4 of the columns of the α i . F or examples of elements α i with α 4 i = 0 , see [5]. 10.2.1 Example Consider α = a + a 7 ∈ Z 2 C 8 . Then α 4 i = 0 and α gener ates an (8 , 6 , 2) linear cyclic code – this is the bes t distance for a linear (8 , 6) co de. Now construc t con volutional co des similar to the construction o f the (2 , 1) co des. An element α 0 ∈ Z 2 C 8 such that α 4 0 = 1 is needed. There ar e a num ber of choices including α 0 = 1 , α 0 = 1 + a + a 3 , α 0 = 1 + a + a 7 . Cho ose α 0 so tha t the first 3 rows of the ma trix of α 0 generates a linear co de of larg est distance. It is ea sy to v erify that the first three rows of α 0 = 1 + a + a 3 generates a linear co de of distance 2. • w = α + α 0 z . This gives a (8 , 6) co de of free distance 4. The ‘degr e e ’ in the conv olutional sense is 6. • w = α + α 0 z + αz 2 . This is a (8 , 6) conv olutiona l co de of free distance 6. The ‘degree’ here is 12. • w = α + α 0 z + αz 2 + αz 3 gives an (8 , 6) co de of free dis ta nce 6. • w = α + α 0 z + αz 3 + αz 4 gives an (8 , 6) co de of free dis ta nce 8. • Polynomial degree 5: w = α + α 0 z + αz 3 + αz 4 + αz 5 . The free distance has to b e determined. • Polynomial degree 6: w = α + αz 2 + αz 3 + α 0 z 4 + αz 5 + αz 6 . This should give a free distance of at least 1 0. • As for the (2 , 1) convolutional co des in Section 7, b y mimicking the p olynomials us e d to generate cyclic co des, it s hould be p ossible to g et (8 , 6 ) con volutional codes with increasing free distance. 11 Using idemp oten ts to generate con vol utional co des Let F G b e the group ring ov er a field F . F or mo st cases in applications it is requir ed that char F 6 | | G | . It may also b e necessary to require that F contains a primitive n th ro ot of unity . The complex num b ers F = C satisfies these conditions. The reader is (again) referred to [9] for background definitions and results on gr o up r ing s in relation to this section. Let { e 1 , e 2 , . . . , e k } be a complete family of orthog onal idemp oten ts in F G . Such sets alw ays exis t when char F 6 | | G | . Thu s: (i) e i 6 = 0 and e 2 i = e i , 1 ≤ i ≤ k . (ii) If i 6 = j then e i e j = 0. (iii) 1 = e 1 + e 2 + . . . + e k . Here 1 is used for the identit y o f F G . Theorem 11.1 L et f ( z ) = k X i =0 ± e i z t i . Then f ( z ) f ( z − 1 ) = 1 . Pro of: Since e 1 , e 2 , . . . , e k is a set of or thogonal primitive ide mp otent s, f ( z ) f ( z − 1 ) = e 2 1 + e 2 2 + . . . + e 2 k = 1.  15 The result in Theorem 11.1 can be c o nsidered a s an identit y in RC ∞ wherein R = F G is a gro up ring. T o now construct convolutional co des, decide on the rank r a nd then use the first r rows o f the matrices of the e i in Theorem 1 1.1. The con trol matr ix is obtained from f ( z − 1 ) b y deleting the last r columns of the e i . If the e i hav e rank ≥ k and fo r some i r a nk e i = k then it is pro bably best to take the r = k for the ra nk of the conv olutional co de, althoug h other cases a lso ha ve uses dep ending o n the a pplication in mind. 11.1 Idemp oten ts in group rings Orthogo nal sets of idempo ten ts ma y b e obtained in group r ing s from the conjugacy clas ses and c haracter tables, see e.g. [9]. Notice also that a pro duct h ( z ) = Q i f i ( z ) where the f i ( z ) satisfy the conditions of Theore m 11.1 also satisfies h ( z ) ˆ h ( z − 1 ) = 1, wher e ˆ h ( z − 1 ) is the pro duct of the f i ( z − 1 ) in r ev erse order , a nd th us h ( z ) can then b e used to define co n volutional codes. In the ring o f matrices define e ii to b e the matrix with 1 in the i th diagonal and zeros elsewhere. Then e 11 , e 22 , . . . , e nn is a complete set of orthog onal idempotents and can b e used to define such f ( z ). These in a sense are trivial but can b e useful and can als o be combined with others . T o construct co n volutional co des: • Find sets of o rthogonal idempotents. • Decide on the f ( z ) to b e used with each set. • T ake the pro duct of the f ( z ). • Decide on the r ate. • Conv ert these idemp oten ts in to matrices a s p er the iso morphism b et ween the gro up ring and a ring of matrices. Group r ing s are a r ic h sour c e of complete sets of or tho gonal idemp oten ts. This br ings us into character theory in gro up r ing s. Orthogona l sets ov er the rationals a nd other fields a re also obtainable. The Computer Algebra pack ages GAP and Mag ma can cons truct character tables and conjugacy classes from which complete se ts of orthogo nal idemp oten ts may be obtained. 11.2 Example 1 Consider C C 2 where C 2 is g enerated b y a . Define e 1 = 1 2 (1 + a ) a nd e 2 = 1 − e 1 = 1 2 (1 − a ). This g iv es f ( z ) = e 1 + e 2 z t or f ( z ) = e 2 + e 1 z t for v arious t . Pr o ducts of these co uld also b e used but in this case we get another of the same for m b y a p o wer of z . 11.3 Cyclic The o rthogonal idemp oten ts a nd character table of the cyclic gr oup a re well-kno wn and are closely related to the F our ier matrix. This gives for example in C 4 , e 1 = 1 4 (1 + a + a 2 + a 3 ) , e 2 = 1 4 (1 + ω a + ω 2 a 2 + ω 3 a 3 ) , e 3 = 1 4 (1 − a + a 2 − a 3 ) , e 4 = 1 4 (1 + ω 3 a + ω 2 a 2 + ω a 3 ) fr om w hich 4 × 4 matr ices with degree 4 in z may b e cons tructed, where ω is a pr imitiv e 4 th ro ot of unity . Notice in this case that ω 2 = − 1. Let f ( z ) = e 1 + e 2 z + e 3 + e 4 z 3 . The n f ( z ) f ( z − 1 ) = 1. W e take the first row of the matrices to give the following gener ator ma tr ix for a (4 , 1 , 3) conv olutional co de: G ( z ) = 1 4 { (1 , 1 , 1 , 1) + (1 , ω , − 1 , − ω ) z + (1 , − 1 , 1 , − 1 ) z 2 + (1 , − ω , − 1 , ω ) z 3 } . It is easy to chec k that a co mbination of a ny one, tw o or three of the vectors (1 , 1 , 1 , 1) , (1 , ω , − 1 , − ω ) , (1 , − 1 , 1 , − 1) , (1 , − ω , − 1 , ω ), which are the rows o f the F ourier matrix, has dis- tance a t least 2 and a combination of all four of them has distance 1. F r om this it is easy to show that 16 the co de has free distanc e 14 – a n y co m bination o f mor e than one will have 4 at ea c h end and thre e in the middle with distance a t least 2. This gives a (4 , 1 , 3 , 14) co n volutional co des which is optimal – see [11]. W e can combine the e i to get real sets of o rthogonal idemp oten ts. Note that it is enough to co m bine the conjuga cy classes of g and g − 1 in or der to get rea l sets of o rthogonal idempotents. In this c a se then we get ˆ e 1 = e 1 = 1 4 (1 + a + a 2 + a 3 ) , ˆ e 2 = e 2 + e 4 = 1 2 (1 − a 2 ) , ˆ e 3 = e 3 = 1 4 (1 − a + a 2 − a 3 ), which can then be used to co nstruct real convolutional co des. Then G ( z ) = 1 4 { (1 , 1 , 1 , 1) + 2(2 , 0 , − 2 , 0) z + (1 , − 1 , 1 , − 1) z 2 } gives a (4 , 1 , 2) convolutional code. Its free distance is 1 0 whic h is also o ptimal. Using C 2 × C 2 gives different matrices. Here the s et of orthogo nal idemp oten ts consists of e 1 = 1 4 (1 + a + b + ab ) , e 2 = 1 4 (1 − a + b − ab ) , e 3 = 1 4 (1 − a − b + ab ) , e 4 = 1 4 (1 + a − b − ab ) and the matrices derived are all real. This gives G ( z ) = 1 4 { (1 , 1 , 1 , 1) + (1 , − 1 , 1 , − 1) z + (1 , − 1 , − 1 , 1) z 2 + (1 , 1 , − 1 , − 1) z 3 } . Its free dista nc e also s e e ms to be 14. 11.4 Symmetric group The o r thogonal idempotents of the symmetric gro up ar e w ell-understo o d a nd are real. W e present an example here from S 3 , the symmetric group o n 3 letters. Now S 3 = { 1 , (1 , 2) , (1 , 3) , (2 , 3) , (1 , 2 , 3) , (1 , 3 , 2) } where these are cycles . W e also use this listing of S 3 when co nstructing matrices. There ar e thre e co nj uga cy cla sses: K 1 = { 1 } ; K 2 = { (1 , 2) , (1 , 3) , (2 , 3) } ; K 3 = { (1 , 2 , 3) , (1 , 3 , 2 ) } . Define ˆ e 1 = 1 + (1 , 2) + (1 , 3) + (2 , 3) + (1 , 2 , 3) + (1 , 3 , 2), ˆ e 2 = 1 − { (1 , 2) + (1 , 3) + (2 , 3) } + (1 , 2 , 3) + (1 , 3 , 2), ˆ e 3 = 2 − { (1 , 2 , 3) + (1 , 3 , 2) } , and e 1 = 1 6 ˆ e 1 ; e 2 = 1 6 ˆ e 2 ; e 3 = 1 3 ˆ e 3 . Then { e 1 , e 2 , e 3 } for m a complete orthogonal set of idemp oten ts and may be used to co nstruct conv olutional co des. The G -matrix of S 3 (see [4]) is         1 (12) (13) (23) (123) (132) (12) 1 (132) (123) (23) (13) (13) (123) 1 (132) (12) (23) (23) (132) (123) 1 (13) (12) (132) (23) (12) (13) 1 (123 ) (123) (13) (23) (21) (132 ) 1         . Thu s the matr ices of e 1 , e 2 , e 3 are r espectively E 1 = 1 6         1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1         E 2 = 1 6         1 − 1 − 1 − 1 1 1 − 1 1 1 1 − 1 − 1 − 1 1 1 1 − 1 − 1 − 1 1 1 1 − 1 − 1 1 − 1 − 1 − 1 1 1 1 − 1 − 1 − 1 1 1         17 E 3 = 1 3         2 0 0 0 − 1 − 1 0 2 − 1 − 1 0 0 0 − 1 2 − 1 0 0 0 − 1 − 1 2 0 0 − 1 0 0 0 2 − 1 − 1 0 0 0 − 1 2         Note that e 1 , e 2 hav e rank 1 and that e 3 has ra nk 2. 12 Other c haracteristics Conv olutional co des ov er fields of ar bitrary characteristic, and no t just characteristic 2, may also b e constructed using the g eneral metho d as previo usly described. The following theorem is similar to Theo rem 5.2. Theorem 12.1 L et R = F G b e the gr oup ring of a gr oup G over a field F with char acteristic p . Su p p ose α i ∈ R c ommute and γ i ∈ F . L et w = n X i =0 α i γ i z i ∈ R C ∞ . Then w p = γ p t z pt if and only if α p i = 0 , i 6 = t and α p t = 1 . The situation with γ t = 1 is easiest to dea l with and is not a gr eat r estriction. Construct conv olutional co des as follows. Find elemen ts α i with α p i = 0 and units u with u p = 1 in the group ring R . Then define elemen ts as in Theorem 12 .1 in R [ z ] to fo rm units in R [ z ]. Thus get f ( z ) p = γ p t z pt and hence f ( z ) × f ( z ) p − 1 / ( γ p t z pt ) = 1 . F r o m these units, conv olutional co des ar e defined as descr ibed in Section 2 or Section 2.2. Thu s f ( z ) may b e used to define a co n volutional co de. By choosing the first r r ows of the α i considered as matrices defines a ( n, r ) co n volutional co de where n = | G | . The generator matrix is ˆ f ( z ) = n X i =0 ˆ α i γ i z i where ˆ α i denotes the firs t r rows of the matr ix of α i . It is necess a ry to decide which ro ws of the matrix to choo se in defining the co nvolutional co de. This is usually decided by co ns idering the rank(s) o f the non-zero α i . 12.1 Examples for ch aracteristic 3 Suppo se then F has c haracter istic 3 a nd co nsider F ( C 3 × C 3 ) where the C 3 are generated resp ectiv ely by g , h . Define α = 1 + h (1 + g ). Then α 3 = 0. Define α 0 = 2 + 2 h . Then α 3 0 = 1. The matrix o f α is P =   I B 0 0 I B B 0 I   where I is the iden tity 3 × 3 matrix , 0 is the zero 3 × 3 matrix and B =   1 1 0 0 1 1 1 0 1   . By r o w (blo c k) op erations P is equiv alent to   I 0 − B 2 0 I B 0 0 0   . Thus P ha s rank 6 and the matrix Q =  I 0 − B 2 0 I B  defines a blo c k (9 , 6) co de which indeed has distance 3. Now define α t = α 0 for some 0 < t < n a nd c ho ose α i = 0 or α i = α fo r i 6 = t . Define f ( z ) = n X i =0 α i z i . Then by T heo rem 12.1, f ( z ) 3 = z 3 t and hence f ( z ) × ( f ( z ) 2 /z 3 t ) = 1. Thus f ( z ) may be used to define a conv olutiona l co de. Cho ose the first 6 r o ws of the α i in f ( z ) to define the co de and th us we get a 18 (9 , 6 ) co n volutional co de. The generator matrix is ˆ f ( z ) = n X i =0 ˆ α i z i where ˆ α i denotes the first 6 rows of α i , considere d as a matrix. The control matrix is obtained fr o m f ( z ) 2 /z 3 t using the last 3 columns of the α i . Lemma 12.1 x ˆ α i + y ˆ α 0 has distanc e at le ast 1 for 1 × 6 ve ctors x , y with y 6 = 0 . 12.1.1 Sp ecific exampl e s for c haracteristic 3 Define f ( z ) = α + α 0 z + αz 2 . Then ˆ f ( z ) = ˆ α + ˆ α 0 z + ˆ αz 2 is a conv olutional (9 , 6) co de of free dis ta nce 8 . Define f ( z ) = α + α 0 z + αz 3 + αz 4 . The n ˆ f ( z ) = ˆ α + ˆ α 0 z + ˆ αz 3 + ˆ αz 4 defines a (9 , 6) con volutional co de which has free dista nce 11. A result similar to Theorem 7.3 ca n also be proved. Suppo se now C is a cyclic ( n, k , d 1 ) code o ver the field F of c hara cteristic 3. Suppose also tha t the dual of C , denoted ˆ C , is an ( n, n − k , d 2 ) co de. Let d = min ( d 1 , d 2 ). Suppo se f ( g ) = r X i =0 β i g i , with β i ∈ F, ( β r 6 = 0), is a generating p olynomial for C . In f ( g ), assume β 0 6 = 0. Consider f ( z ) = r X i =1 α i z i where no w α i = β i α with α as ab ov e in F ( C 3 × C 3 ). No te that if β i = 0 then α i = 0. Replace so me α i , say α t , by α 0 (considered as mem b ers o f F ( C 3 × C 3 )). So assume f ( z ) = r X i =0 α i z i with this α t = α 0 and other α i = β i α (for i 6 = t ). Then f ( z ) 3 = β 3 t z 3 t giving that f ( z ) × ( f ( z ) 2 / ( β 3 t z 3 t ) = 1. W e now use f ( z ) to gener a te a c o n volu- tional co de by ta king the firs t 6 rows of the α i . Thus the genera tin g matrix is ˆ f ( z ) = r X i =0 ˆ α i β i z i where ˆ α i consists o f the first 6 rows of α for i 6 = t and ˆ α t consists of the fir s t 6 rows of α 0 . F or the following theore m assume the inv ertible element α 0 do es not o ccur in the fir s t or the last po sition of f . Theorem 12.2 L et C denote t h e c onvolutional c o de with gener ator matrix ˆ f . Then the fr e e distanc e o f C is at le ast d + 4 . 13 General considerations Suppo se it is required that a de g ree n p olynomial f ( z ) = α 0 + α 1 z + α 2 z 2 + . . . + α n z n is to ha ve an inv erse in R [ z ]. Then sufficient conditions on the α i are obtained by formally mult iplying f ( z ) by a general g ( z ) and making sure in the product that the co efficient of z 0 is 1 and the coefficient of z i is 0 for i > 0 . If all the α i commute (as for g roup rings o n a belian groups), 2 α i = 0 (a s in character istic 2 ), and α 2 0 = 1 , α 2 i = 0 , ∀ i ≥ 1 , then g ( z ) = α 0 − α 1 z − α 2 z 2 − . . . − α n z n satisfies f ( z ) g ( z ) = 1. Cho osing different α i will max imise the distance. It is easy to obtain elements α in the g roup ring with α 2 = 0 . Co nsider for example Z 2 C 2 n . Then w i = g i + g n + i for 0 ≤ i < n satisfy w 2 i = 0 a nd any co m bina tion α of the w i satisfies α 2 = 0 . It is then a matter of cho osing suitable combinations. 13.1 Nilp oten t t yp e Many group rings R ha ve element s α suc h that α n = 0 (a nd α r 6 = 0 , r < n ). These can b e e x ploited to pro duce conv olutional co des. 19 13.1.1 Example Consider F C 14 where F ha s characteristic 2. Let w 0 = 1 + g 5 + g 6 + g 12 + g 13 , w 1 = 1 + g 2 + g 5 + g 7 + g 9 + g 12 , w 2 = 1 + g + g 3 + g 7 + g 8 + g 10 and define p = w 0 + w 1 z + w 2 z 2 . Then p 2 = 1. Since w 2 i = 0 for i ≥ 1, consider a rate of 1 2 . Thus consider the con volutional co de with encoder matrix obtained from the firs t 7 rows of p and then the control matrix is obtained fr om the las t 7 columns of p . 13.2 F urther examples Consider Z 2 C 8 generated by g . Define u = α 0 + (1 + g 4 ) z + (1 + g 2 ) z 2 + (1 + g ) z 3 where α 8 0 = 1. There are a num ber of choices for α 0 , e.g. α 0 = 1 + g + g 3 . Then u 2 = α 2 0 + (1 + g 8 ) z + (1 + g 4 ) z 2 + (1 + g 2 ) z 6 = α 2 0 + (1 + g 4 ) z 2 + (1 + g 2 ) z 6 , u 4 = α 4 0 + (1 + g 4 ) z 12 and u 8 = 1. Then u can be used to define a conv olutional co de . Now 1 + g 4 has rank 4 so for b est results mak e it an (8 , 4) conv olutional co de by taking the first 4 rows of the matrices of u . This is an (8 , 4 , 9) con volutional co de with degree/ memory δ = 6. The rate could b e increased but this would r educe the contribution fr om (1 + g 4 ) matrix to distanc e essentially 0 as it has rank = 4. This would g iv e a (8 , 6 , 7) conv olutional co de. T o go further, consider Z 2 C 16 etc. . Here use degree 6 or 3 as the largest p o wer of z and it is then po ssible to get a (16 , 8 , 9) conv olutional co de. As rank(1 + g 4 ) = 8 it is probably p ossible to construct a (16 , 12 , 9 ) but details have not been worked o ut . These ar e binar y codes . Going to bigger fields should give better distances. 14 Hamming t yp e Set R = Z 2 ( C 4 × C 2 ). Suppos e C 4 is genera ted by a a nd C 2 is genera ted by h . Consider α 0 = 1 + h (1 + a 2 ) and α i = 1 + h ( a + a 2 + a 3 ) or α i = 0 for i > 0. Then α 2 0 = 1 and α 2 i = 0. Define w ( z ) = n X i =0 α i z i in RC ∞ . B y Theorem 5.1, w 2 = 1. Let A =     1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1     , B =     0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0     and I is the iden tity 4 × 4 matrix. The matrix corres p onding to α 0 is then  I A A I  and the matrix corr esponding to α i , i 6 = 0, is either  I B B I  or the zero matr ix. Now sp ecify that the first 4 r o ws of w for m ula te the g enerator ma trix of a co de a nd then the last four columns o f w for m ulate the con trol matrix. This g iv es a con volutional c o de o f leng th 8 and dimens io n 4. It is easy to transfor m the resulting co de into a systematic co de. The genera tor matr ix is G ( z ) = ( I , A ) + δ 1 ( I , B ) z + δ 2 ( I , B ) z 2 + . . . + δ n ( I , B ) z n , wher e δ i ∈ { 0 , 1 } . The co n trol matrix is H ( z ) =  A I  + δ 1  B I  z + δ 2  B I  z 2 . . . + δ n  B I  z n . The ( I , A ) may be mo ved to the co efficien t o f any z i in which case the (natural) control matrix will need to b e divided by a power of z to get the true control matrix. This convolutional co de may be considered as a Hamming t yp e conv olutional c o de as ( I , B ) is a generator matrix of the Hamming (8 , 4) co de. F or n = 1 the free distance turns out to be 6; this can be prov ed in a s imilar manner to Theor em 9.1. 20 14.1 Example of this type G ( z ) = ( I , B ) + ( I , A ) z + ( I , B ) z 2 with control matrix H ( z ) /z 2 where H ( z ) =  B I  +  A I  z +  B I  z 2 has free distance 10. 14.2 F rom c yc lic t o Hamming type F or n ≥ 2, pro ceed as pr eviously to define the p olynomials b y reference to corresp onding cyc lic linear po lynomials. This will give conv olutional code s o f this t yp e of increasing free distance. No te that ( I , A ) has distance 2 , ( I , B ) (the Hamming Co de) has dis tance 4, a ny co m bina tion of ( I , A ) and ( I , B ) has distance ≥ 1 . The following may be proved in a simila r manner to Theo rem 7.3. Suppo se now C is a cyclic ( n, k, d 1 ) co de over the field F of characteris tic 2 and that the dual of C , ˆ C , is an ( n, n − k , d 2 ) co de. Let d = min ( d 1 , d 2 ). Assume f ( g ) = r X i =1 β i g i is a genera tor p olynomial for C . In f ( g ), it is poss ible to arrange that β 0 6 = 0 and naturally assume that β r 6 = 0 . Define f ( z ) = r X i =1 α i z i with the α i = β i α i , i 6 = t and α t = α 0 . Then f ( z ) 2 = z 2 t giving f ( z ) × f ( z ) /z 2 t = 1 . Now use f ( z ) to g enerate a conv olutional co de by taking just the first four rows o f the α i . Thus the genera ting matrix is G = r X i =0 ˆ α i z i where ˆ α i consists of the firs t four rows of the matrix of α i . Theorem 14.1 C has fr e e distanc e at le ast d + 8 . References [1] Richard E. Blahut, A lgebr aic Co des f or da ta tr ansmission , Cambridge Univ ers ity Pres s , 2 0 03. [2] Gluesing-Luerss en, Heide & Sc hmale, Wiland, “ On Cyclic Con volutional Co des”, Acta Applicandae Mathematicae, V ol. 82 , No. 2, 2 0 04, 183-237 . [3] Paul Hurley and T ed Hurley , “Co des from zer o-divisors and units in g roup ring s”, arXiv:07 10.5893 . [4] T ed Hurley , “ Group r ings and r ing s of matrices”, Inter. J. P ur e & Appl. Ma th., 31 , no .3, 2 006, 319-3 35. [5] T ed Hurley , “Self-dua l, dua l-con taining and related q uan tum co des from group rings”, preprint av ailable at http: [6] Paul Hurley and T ed Hurley , “ Mo dule co des in group r ings”, ISIT2007, Nice, 1981 - 1985, 2007 . [7] v an Lint , J.H. and Wilson, R.M., A c ourse in Combinatori cs , Cam bridge Universit y P ress, 2001. [8] R. J. McEliece, “The alg ebraic theory of co n volutional co des”, in Handb o ok of Co ding The ory, V olume I , North Holland, Elsev ie r Scie nce, 1998. [9] C´ esar Milies & Sudarsha n Sehga l, A n intro duct i on to Gr oup R ings , Klumer , 2002. [10] David J . C. MacK a y , Information The ory, Infer enc e and L e arning Algorithms , Cambridge University Press, 20 03. [11] J. Ro s en thal & R. Smarandache, “ Maximum distance separable conv olutiona l co des”, Appl. Algebra Engrg. Comm. Comput. 10 (1), 15-3 7, 1999. 21 [12] R. Smarandache, H. Gluesing-Luer ssen, J. Rosenthal, “C o nstructions for MDS-co n volutional co des”, IEEE T rans. Inform. Theory , vol. IT-47, 2 045-2049 , 200 1 . [13] F.J.MacWilliams and N.J.A. Sloane, The The ory of Err or-Corr e ct i ng Co des , North Ho lland, 1977. National University o f Irela nd, Galw ay Galwa y Ireland. 22