On minimality of convolutional ring encoders

1 On minimalit y of con v olutional ring encoders Mar greta Kuijper and Raquel Pinto Abstract —Con v olutional codes are c onsidered with code se- quences modelled as semi-infini te Laur ent series. It is wellknown that a con vo lutional code C ov er a finit e gro up G has a min imal trellis representation that can b e deriv ed from code sequences. It is also wellknown that, f or the case th at G is a fin ite fi eld, any polynomial encoder of C can be algebraically manipulated to yield a min imal polynomial encoder wh ose controller canonical realization is a minimal trellis. In this paper we seek to extend this result to the finite ring case G = Z p r by introducing a socalled “ p -encoder”. W e show ho w to manipu late a polynomial encoding of a noncatastrophic con v olutional code ov er Z p r to produce a particul ar t ype of p -encoder (“minimal p -encoder”) whose controller canonical realization is a minimal trellis with nonlinear features. The minimum number of trellis states is then expressed as p γ , wh ere γ is the sum of the ro w degrees of the minimal p -encoder . In particular , we show that any con vo lutional code ov er Z p r admits a delay-free p -encoder which implies the nov el result that delay-freeness is not a pro perty of the code but of the encoder , just a s in t he field case . W e conjecture that a similar result hold s with respect to catastr ophicity , i.e., any catastrophic con v olutional code o ver Z p r admits a noncatastr ophic p -encoder . I . I N T R O D U C T I O N There exists a considera ble b ody of literature on conv olu- tional codes over fin ite groups. In this paper we are interested in trellis r epresentation s that u se a min imum n umber of states. Since decoders, s uch as the V iterbi decoder, are b ased on trellis representatio ns, m inimality is a desirable p roperty that le ads to low complexity decodin g. In [6, Sect. VI -D] a m inimal encoder construction is presented in ter ms o f code sequences of the code, in volving soca lled “gran ule representa ti ves”, see also [16]. This is a p owerful metho d tha t applies to con- volutional cod es over any fin ite g roup G . It is wellkn own that, fo r the case that G is a field, any polyno mial encoder of a convolutional co de c an be algeb raically man ipulated to y ield a so-called “can onical polyn omial encod er” ( left prime an d r ow reduced) whose contr oller canonical realization yields a m inimal trellis rep resentation of th e code. This is a fundam ental result that is usefu l in pra ctice bec ause co des are usually specified in ter ms of encod ers rather than code sequences. In th is p aper we seek to exten d th is result to the finite ring case G = Z p r , where r is a positive integer and p is a prime integer . The o pen prob lem that we solve is also mentioned in the 2007 paper [23]. W e first tailo r the concept of encoder to the Z p r case, m aking use of the spec ific alg ebraic finite ch ain structure of Z p r . T his lead s to con cepts of “ p - encoder ” and “minimal p -en coder” . W e then show h ow to construct a m inimal p -enc oder from a p olyno mial encoding M. Kuij per is with the Department of EE Enginee ring, Uni versity of Melbourne , VIC 3010, Australi a E-m ail: m.kuijper@ee.u nimelb.edu.au . R. Pinto is with the Department of Mathema tics, Univ ersity of A veiro, 3810-193 A veiro , Portugal E-ma il: raquel@ua.pt . of the co de. The minim al p - encoder translates immed iately into a minim al tr ellis realization. Th us our results allow for easy constructio n of a minimal trellis representation from a polyno mial encodin g an d parallel the field case. Con volutional co des over r ings were intro duced in [17], [1 8] where th ey are mo tiv ated fo r use with phase mo dulation . I n particular, con volutional codes over the ring Z M are useful for M -ary phase modulation (with M a positive integer). By the Chinese Remainder Th eorem, results on code s over Z p r can be extended to codes over Z M , see also [ 19], [1], [2], [9]. Most of the literature o n conv olutional codes over rings adop ts an appr oach in which code sequences are semi-infinite Lau rent series [6] , [21], [15], [16], [9], [3], [27], [ 26]. In ord er to make a con nection with th is liter ature, we a dopt th is appr oach in our definition of a conv olu tional code: a linear conv olutional co de C o f length n o ver Z p r is defin ed a s a s ubset of ( Z n p r ) Z for which there exists a polynom ial matrix G ( z ) ∈ Z k × n p r [ z ] , such that C = { c ∈ ( Z n p r ) Z | ∃ u ∈ ( Z k p r ) Z : c = u G ( z ) and supp u ⊂ [ N , ∞ ) for so me integer N } . (1) Here supp u d enotes the suppo rt o f u , i.e., the s et of time- instants t ∈ Z fo r which u ( t ) is non zero. Further, z denotes the right shift operator z u ( t ) = u ( t − 1) . Clearly , ( 1) implies that C is linear and shift- in variant with respect to b oth z and z − 1 . If the matrix G ( z ) has full row rank then G ( z ) is called an encoder of C . For the field case any linear co n volution al c ode admits a left prime polynomial en coder, i.e., an encoder that has a polyno mial righ t inverse. Such an encod er G ( z ) gives rise to the following two p roperties: 1) dela y-fr ee pr ope rty : fo r any N ∈ Z supp c ⊂ [ N , ∞ ) = ⇒ supp u ⊂ [ N , ∞ ) 2) no ncatastr op hic pr op erty : supp c is finite = ⇒ supp u is finite , where c = u G ( z ) . Clearly , in the field case, “delay -free”- ness and “ catastrophicity ” ar e encoder prope rties, n ot co de proper ties. For the ring case, howe ver , there are codes that do not ad mit a nonc atastrophic enco der . For exam ple ( see [6], [21], [4]) the co n volutional co de over Z 4 with encoder G ( z ) = [1 + z 1 + 3 z ] does no t admit a no ncatastrop hic encod er . Similarly , the rotationally in variant co n volution al co de over Z 4 with encoder G ( z ) =  3 + 3 z + 3 z 2 3 + z + z 2  does not admit a no ncatastrop hic encoder . The read er is referr ed to [18] for motiv atio n and character ization of rotatio nally inv aria nt codes ov er rings. Further, th ere are codes that do not admit a delay-fr ee e ncoder . For example (see [18], [16], [ 4]) the 2 conv olutio nal co de over Z 4 with encod er G ( z ) = [2 2 + z ] does no t ad mit a delay-f ree encoder . Note that so me cod es over Z p r do n ot even admit an encoder, f or examp le over Z 4 the code giv e n by (1) with G ( z ) =  1 + z z z 2 2 2 2  . The literature (see e.g. [4, subsect. V -C]) has declared the proper ties of “delay-fr ee” and “ca tastrophic” to be pro perties of the co de rather than the e ncoding procedu re. B y resorting to a pa rticular typ e of polynom ial encoder, named “ p -encod er”, we show in section III that dela y-freen ess is not a p roperty of the code but of the encoding procedu re, just as in the field case, see also [12]. W e conjecture that the same is tr ue for catastrophicity . T o supp ort this argument, in section IV we examine specific catastrophic co n volutional co des over Z p r and show that a nonca tastrophic p -encode r exists f or these examples. A m ore recent app roach [22] ( see also [7], [23]) to conv o- lutional codes f ocuses on so-called “fin ite sup port con volu- tional codes” in which the input sequen ce u correspond s to a polyno mial. Th us the natural time axis is Z + and both input sequenc es and code sequen ces have finite suppo rt. Finite support conv olutional codes are, by definition , noncatastrop hic (Property 2 above) an d can be interpreted as submodules of Z n p r [ z ] . For n = 1 connectio ns can be made with polyn omial block codes. For more details the reader is refe rred to our paper [11]. I I . P R E L I M I N A R I E S A set that plays a funda mental role throug hout th e pap er is the set of “digits”, de noted by A p = { 0 , 1 , . . . , p − 1 } ⊂ Z p r . Recall that any element a ∈ Z p r can be written un iquely as a = θ 0 + θ 1 p + · · · + θ r − 1 p r − 1 , where θ ℓ ∈ A p for ℓ = 0 , . . . , r − 1 ( p -ad ic exp ansion). This fu ndamen tal proper ty of the ring Z p r essentially expresses a typ e of linea r indepen dence among the elements 1 , p , p 2 , ... , p r − 1 . It leads to specific notio ns o f “ p -linear indepen dence” and “ p -gener ator sequence” for mod ules in Z n p r , a s developed in th e 1 996 paper [24]. For example, f or the simplest case n = 1 , the elements 1 , p , p 2 , ... , p r − 1 are ca lled “ p - linearly in depend ent” in [24] and the module Z p r = span { 1 } is written as Z p r = p − span { 1 , p, p 2 , . . . , p r − 1 } . The mo dule Z p r is said to hav e “ p -dimension ” r . In th is section we recall the main concepts fro m [13] on mod- ules in Z n p r [ z ] , tha t are needed in th e sequ el. W e p resent the notions of p -basis and p -d imension o f a submodule o f Z n p r [ z ] , which are extensions fro m [24]’ s notions for submodu les of Z n p r . From [ 13] we also recall the concept of a red uced p -ba sis in Z n p r [ z ] that pla ys a cru cial r ole in the next section. Definition II.1. [13] Le t { v 1 ( z ) , . . . , v m ( z ) } ⊂ Z n p r [ z ] . A p -linea r co mbination o f v 1 ( z ) , . . . , v m ( z ) is a vecto r m X j =1 a j ( z ) v j ( z ) , whe re a j ( z ) ∈ Z p r [ z ] is a polynomial with coefficients in A p for j = 1 , . . . , m . Furthe rmore, the set of all p - linear co mbination s of v 1 ( z ) , . . . , v m ( z ) is deno ted by p -span ( v 1 ( z ) , . . . , v m ( z )) , whereas the set of all linear combinatio ns of v 1 ( z ) , . . . , v m ( z ) with coefficients in Z p r [ z ] is denoted b y spa n ( v 1 ( z ) , . . . , v m ( z )) . Definition II.2. [13] A sequence ( v 1 ( z ) , . . . , v m ( z )) of vectors in Z n p r [ z ] is said to be a p -genera tor sequence if p v m ( z ) = 0 and p v i ( z ) is a p -linear combinatio n of v i +1 ( z ) , . . . , v m ( z ) for i = 1 , . . . , m − 1 . The n ext lemma is a straigh tforward result th at is u sed in section III. Lemma II.3. Let ( v 1 ( z ) , . . . , v m ( z )) be a p -generator se- quence in Z n p r [ z ] . Then ( v 1 (0) , . . . , v m (0)) is a p -generator sequence in Z n p r . Theorem II.4. [13] Let v 1 ( z ) , . . . , v m ( z ) ∈ Z n p r [ z ] . If ( v 1 ( z ) , . . . , v m ( z )) is a p -generator sequen ce th en p − span ( v 1 ( z ) , . . . , v m ( z )) = spa n ( v 1 ( z ) , . . . , v m ( z )) . In pa rticular , p − span ( v 1 ( z ) , . . . , v m ( z )) is a submodule of Z n p r [ z ] . Definition II.5. [13] The vectors v 1 ( z ) , . . . , v m ( z ) ∈ Z n p r [ z ] are said to be p - linearly independent if the only p -lin ear combinatio n o f v 1 ( z ) , . . . , v m ( z ) that equ als zero is the tri v ial one. Definition II.6. L et M be a submodu le of Z n p r [ z ] , wr itten as a p -span o f a p -genera tor sequence ( v 1 ( z ) , v 2 ( z ) , · · · , v m ( z )) . Then ( v 1 ( z ) , v 2 ( z ) , · · · , v m ( z )) is called a p -basis fo r M if the vectors v 1 ( z ) , . . . , v m ( z ) are p -lin early in depend ent in Z n p r [ z ] . Lemma II.7. [13] Let M be a submodule o f Z n p r [ z ] and let ( v 1 ( z ) , v 2 ( z ) , · · · , v m ( z )) b e a p -basis for M . Then each vec- tor of M is written in a u nique way as a p - linear comb ination of v 1 ( z ) , . . . , v m ( z ) . All submodu les o f Z n p r [ z ] can b e written as the p -span of a p - generato r sequen ce. In fact, if M = span ( g 1 ( z ) , . . . , g k ( z )) then M is the p -span of the p - generato r sequence ( g 1 ( z ) , pg 1 ( z ) , . . . , p r − 1 g 1 ( z ) , . . . , g k ( z ) , . . . , p r − 1 g k ( z )) . Next, we recall a particular p - basis for a subm odule of Z n p r [ z ] , called “red uced p -basis”. W e first recall the concept of “degree” of a vector in Z n p r [ z ] , which is the same as in the field case. Definition II.8. Let v ( z ) be a nonzero vector in Z n p r [ z ] , written as v ( z ) = v 0 + v 1 z + · · · + v d z d , with v i ∈ Z n p r , i = 0 , . . . , d , and v d 6 = 0 . Then v ( z ) is said to have degree d , de noted by deg v ( z ) = d . Furtherm ore, v d is called th e leading coefficient vecto r of v ( z ) , denoted by v lc . In the sequel, we denote the leading r ow coefficient matrix o f a polyno mial matrix V ( z ) by V lr c . A matrix V ( z ) is called row-reduced if V lr c has f ull row rank. 3 Lemma II.9. [13] Let M be a submodule of Z n p r [ z ] , written as a p -spa n of a p -generator seque nce ( v 1 ( z ) , . . . , v m ( z )) with v lc 1 , . . . , v lc m p -linearly indepen dent in Z n p r . Then ( v 1 ( z ) , . . . , v m ( z )) is a p -basis for M . Definition II.10 . [13] Let M be a submo dule of Z n p r [ z ] , writ- ten as a p -span of a p -gen erator sequen ce ( v 1 ( z ) , . . . , v m ( z )) . Then ( v 1 ( z ) , . . . , v m ( z )) is called a reduced p -basis fo r M if the vectors v lc 1 , . . . , v lc m are p -linearly independe nt in Z n p r . A reduced p -basis in Z n p r [ z ] generalizes the concept of r ow reduced basis from the field case. Mo reover , it also leads to the predictab le degree pro perty and gives rise to se veral in variants of M , see [13]. In particular, the num ber of vectors in a redu ced p -basis as well as the degrees of th ese vector s (called p -degrees ), are in variants o f M . Consequently , th eir sum is also an inv arian t of M . Every submod ule M of Z n p r [ z ] has a redu ced p -b asis. A constructive p roof is gi ven by Algorith m 3.11 in [13] that takes as its input a set of spann ing vectors and prod uces a reduced p - basis of M . It is easy to see that if the inpu t is already a p -basis of m vectors, then the algo rithm produ ces a reduced p -basis o f again m vector s. Since m is an in variant of the mod ule, it follows that all p -bases of M have the same number of eleme nts. As a result, the next definition is well- defined and no t in co nflict with the slightly different defin ition of [13]. Definition II.11. Th e n umber o f elements of a p -basis o f a submodu le M of Z n p r [ z ] is called th e p -dimension of M , denoted as p − dim ( M ) . In recent work [ 14] it is shown that co mputation al packages for compu ting minimal Gr ¨ obner bases can b e used to con struct a minimal p -encoder . I I I . M I N I M A L T R E L L I S C O N S T RU C T I O N F RO M A p - E N C O D E R Formally , we de fine a tr ellis section as a three-tu ple X = ( Z n p r , S, K ) , w here S is the tr ellis state set and K is the set of branches which is a subset of S × Z n p r × S , see also [6], [16]. A tr ellis is a seq uence X = { X t } t ∈ Z of trellis sections X t = ( Z n p r , S, K t ) . A path through the tr ellis is a seque nce ( · · · , b t − 1 , b t , b t +1 , · · · ) of branch es b t = ( s t , c t , s t +1 ) ∈ K t such tha t b t +1 starts in the trellis state where b t ends fo r t ∈ Z . The set of all trellis paths that start at the zer o state is den oted by π ( X ) . The mapping λ : π ( X ) 7→ ( Z n p r ) Z assigns to every path ( · · · , b t − 1 , b t , b t +1 , · · · ) its label se- quence ( · · · , c t − 1 , c t , c t +1 , · · · ) . A tre llis X is called a tr ellis r epresentation for a conv olutio nal code C if C = λ ( π ( X )) . A trellis representation X for a con volutional code C is called minimal if the size o f its trellis state set S is minimal amo ng all trellis representations of C . It is wellknown how to construct a minimal trellis r epresentation in term s of the cod e sequences of C . In fact, the theo ry o f cano nical trellis rep resentations from the field case carries through to the ring ca se, see [25], [6], [16]. Since it plays a crucial r ole in the proof o f our main result, we recall the definition o f canon ical trellis in Appendix A. Let u s recall the wellknown contro ller canonical for m. Let R be a ring. A matrix E ( z ) ∈ R κ × n [ z ] is realized in controller canonical form [10] (see also [5, Sect. 5]) as E ( z ) = B ( z − 1 I − A ) − 1 C + D , (2) as follows. Denoting the i ’th row o f E ( z ) by e i ( z ) = P δ i ℓ =0 e i,ℓ z ℓ , where e i,ℓ ∈ R 1 × n and e i,δ i 6 = 0 , the matrices A , B , C and D in (2) are gi ven b y A =    A 1 . . . A κ    , B =    B 1 . . . B κ    , C =    C 1 . . . C κ    , D =    e 1 , 0 . . . e κ, 0    , where A i is a δ i × δ i matrix, B i is a 1 × δ i matrix and C i is a δ i × 1 m atrix, g iv en by A i =       0 1 . . . . . . . . . 1 0       , B i =  1 0 · · · 0  , C i =    e i, 1 . . . e i,δ i    for i = 1 , . . . , κ. (3) Whenever δ i = 0 , the i th blo ck in A as well as C is absent and a zer o row occurs in B . Denoting the sum of the δ i ’ s by δ , it is clear that A is a δ × δ nilpotent matrix. The above contr oller canonical realization can be v isualized as a f eedforward shift- register with δ registers. In the c ase that R is a field wit h q elements it is we llknown [8], [16] how to obtain a minimal trellis representatio n for C from a poly nomial encoder . For this, the rows of the poly nomial encoder sho uld first be algeb raically manipu lated (using Smith form and row redu ction operations) to yield a left prime an d row red uced en coder G ( z ) . Th en G ( z ) is called canon ical in the literature, see [16, App . II]. A minim al trellis rep re- sentation of C is then provide d by the contro ller cano nical realization G ( z ) = B ( z − 1 I − A ) − 1 C + D as in (3). Althoug h this r esult is known, in Appe ndix B we give a p roof by showing that there exists an isomorph ism between the tre llis state set of the contr oller canonica l realization and the trellis state set of the canonica l trellis (as defined in Appen dix A) of C . Th e set is thus min imal an d has q ν elements, where q is the number of e lements of the field and ν is the sum of th e row degrees of G ( z ) . The in variant ν is common ly referred to as the “degree” of th e code C (b ut called the “overall constraint length” in the early litera ture). The ro w degrees are called the “Forney indices” of the code [20]. Below we co nsider conv o lutional codes over Z p r that a dmit a no ncatastrop hic enco der, for simp licity , we call such cod es 4 noncatastro phic. W e show that such co des a dmit a particu lar type of po lynom ial encoder (later called “minimal p - encode r”), whose controller can onical realization provide s a m inimal trellis repr esentation, just as in the field case. W e are then also able to express the minim al number of trellis states in terms of the sum of the row degrees of a minimal p - encoder . Let u s now fir st intro duce the no tion of “ p -encoder”. Recall that A p = { 0 , 1 , . . . , p − 1 } ⊂ Z p r . Definition III.1. Let C be a con volutional co de of len gth n over Z p r . Let E ( z ) ∈ Z κ × n p r [ z ] be a polyno mial ma trix whose rows ar e a p -linear ly independe nt p -gen erator sequence. Then E ( z ) is said to be a p -encoder for C if C = { c ∈ ( Z n p r ) Z | ∃ u ∈ ( A κ p ) Z : c = u E ( z ) and supp u ⊂ [ N , ∞ ) f or som e integer N } . The integer κ is called the p -dimension of C . Furthermore , E ( z ) is said to be a delay -free p -enco der if for any N ∈ Z and any c ∈ C , written as c = u E ( z ) with u ∈ ( A κ p ) Z we have supp c ⊂ [ N , ∞ ) = ⇒ supp u ⊂ [ N , ∞ ) . Also, E ( z ) is said to be a noncatastrophic p -encoder if f or any c ∈ C , w ritten as c = u E ( z ) with u ∈ ( A κ p ) Z we hav e supp c is finite = ⇒ supp u is finite . Finally , a conv olu tional code C that admits a non catastrophic p -encod er is called noncatastrophic . Thus a difference between a p -en coder E ( z ) and the en coding matrix G ( z ) of (1), is that the inp uts of E ( z ) take their values in A p rather than in Z p r . Note that the idea of using a p -a dic expansion for the inpu t sequence is already present in the 199 3 paper [6]. I t was not until 199 6 that the crucial notion of p - generato r seq uence ap peared in [24], but only for constant vectors — it was extended to polyn omial vector s in [13]. In our definitio n the rows of a p -encod er are required to be a p -gener ator sequen ce co nsisting o f polyno mial vectors. Recall that a co n volutional code over Z p r is gi ven by (1): C = { c ∈ ( Z n p r ) Z | ∃ u ∈ ( Z k p r ) Z : c = u G ( z ) and supp u ⊂ [ N , ∞ ) f or some integer N } . Also recall that there exist con volution al codes over Z p r that do not admit a G ( z ) of full row ran k, i.e. an enco der . An importan t ob servation is that any co n volutional code over Z p r admits a p -en coder, even a p -e ncoder E ( z ) , suc h th at the rows of E lr c are p - linearly indep endent in Z n p r . In deed, any red uced p -basis of the po lynomial m odule sp anned by the rows of G ( z ) , pro duces the ro ws of such a p -encoder E ( z ) . This shows that the conc ept of p -encoder is more natu ral than the con cept of encod er as it is ta ilored to the algebraic structure of Z p r . The next lemma is straightfo rward. Lemma III.2. Let E ( z ) ∈ Z κ × n p r [ z ] be a p -enc oder for a convolutional cod e C of length n . Then E ( z ) is delay- fr ee pr operty (Definition I II.1) if and only if the r ows of E (0) ar e p -linearly independ ent in Z n p r . Theorem II I.3. Let C be a convolutional code of length n over Z p r . Then C admits a delay-free p -encod er E ( z ) ∈ Z κ × n p r [ z ] for some integer κ , such that the r o ws of E lr c ar e p -linearly indepen dent in Z n p r . Pr oof: As noted above, C a dmits a p -encod er E ( z ) , such that th e rows of E lr c are p -linearly independ ent in Z n p r , i.e., they con stitute a red uced p - basis. W ithout loss of gener ality we may assume that the row degrees of E ( z ) are nonin creasing. Let L be the smallest non negative integer such that the last κ − L rows of E ( z ) are a d elay-free p - encode r . Now assume that L > 0 ( otherwise we are d one). If L = κ it means that the last row e κ ( z ) of E ( z ) can be written as e κ ( z ) = z ℓ ¯ e κ ( z ) , where ℓ > 0 and ¯ e κ ( z ) ∈ Z n p r [ z ] with ¯ e κ (0) 6 = 0 . No te that deg ¯ e κ ( z ) < deg e κ ( z ) . Clearly , ( e 1 ( z ) , . . . , e κ − 1 ( z ) , ¯ e κ ( z )) is a p - encode r of C , whose rows are still a r educed p -basis. If L < κ , the n, by c onstruction , there exist α j ∈ A p for j = L + 1 , . . . , κ , such that e L (0) + X j >L α j e j (0) = 0 (use the fact that ( e 1 (0) , . . . , e κ (0)) is a p - generato r sequence by Lemma II.3). Rep lacing e L ( z ) by ˜ e L ( z ) := e L ( z ) + P j >L α j e j ( z ) obviously gives a p -basis ( e 1 ( z ) , . . . , e L − 1 ( z ) , ˜ e L ( z ) , e L +1 ( z ) , . . . , e κ ( z )) of th e modu le spanned by e 1 ( z ) , . . . , e L ( z ) , . . . , e κ ( z ) and, co nsequen tly , a p -encoder of C . Moreover , by the p -pr edictable degree proper ty (Theo rem 3.8 of [13]), deg ˜ e L ( z ) = deg e L ( z ) , wh ich me ans that ( e 1 ( z ) , . . . , ˜ e L ( z ) , . . . , e κ ( z )) is still a reduced p - basis. Since ˜ e L (0) = 0 , we can write ˜ e L ( z ) = z ˜ ℓ ¯ e L ( z ) , with ¯ e L (0) 6 = 0 and ˜ ℓ > 0 . Note tha t p ˜ e L ( z ) is a p -linear com bination p ˜ e L ( z ) = P j >L β j ( z ) e j ( z ) with β j ( z ) ∈ A p [ z ] . Because of the p - linear ind ependen ce of e L +1 (0) , . . . , e κ (0) , we must have that th e coefficients β j ( z ) are of the form β j ( z ) = z ℓ j ¯ β j ( z ) with ℓ j ≥ ˜ ℓ for L + 1 ≤ j ≤ κ . Con sequently , the seque nce ( e 1 ( z ) , . . . , e L − 1 ( z ) , ¯ e L ( z ) , e L +1 ( z ) , . . . , e κ ( z )) is a p - encoder of C , which is st ill a reduced p -basis with deg ¯ e L ( z ) < deg e L ( z ) . If ( e 1 ( z ) , . . . , ¯ e L ( z ) , e L +1 ( z ) , . . . , e κ ( z )) is not a delay-fr ee p -encod er , the n re-order the vectors so that th eir degrees are non increasing and repeat this proced ure until a delay-fr ee p -enc oder for C is obtain ed. Since the sum of the row degrees of p -bases obtained at each step o f the p rocedu re is lo wer than in th e previous step, a delay- free p -encode r is obtained after fin itely many iteration s. The next example is a simple examp le that illustrates the above theorem . Example III.4. Over Z 4 : c onsider the (2 , 1) conv o lutional code C of [16, p. 1668] g iv en by the polyn omial encod er G ( z ) = [2 2 + z ] . 5 A delay-free p -encoder f or C is given by E ( z ) =  2 2 + z 0 2  . Theorem III.5. Let C be a n oncata str ophic con vo lutional code of len gth n over Z p r . Then C admits a delay- fr ee nonca tastr ophic p -en coder E ( z ) ∈ Z κ × n p r [ z ] for some in te ger κ , such that th e r o ws of E lr c ar e p -linearly indepe ndent in Z n p r . Pr oof: By defin ition there exists a nonc atastrophic p - encoder E 1 ( z ) for C . Apply Algorithm 3.11 of [13] to the ro ws of E 1 ( z ) . This gives us a reduced p -basis e 1 ( z ) , . . . , e κ ( z ) for the mod ule spanned by the rows of E 1 ( z ) . Define E 2 ( z ) as the κ × n polyno mial m atrix with e 1 ( z ) , . . . , e κ ( z ) as rows. By co nstruction the ro ws of E lr c 2 are p -linearly independ ent in Z n p r . I t is easy to see th at E 2 ( z ) is still noncatastroph ic. If E 2 ( z ) is not delay-free apply the p rocedu re of the p roof of Theorem III.3 to E 2 ( z ) to o btain a d elay-free p -encod er E ( z ) , such that the rows of E lr c are p -linearly ind epend ent in Z n p r . It is ea sy to see th at E ( z ) is still noncatastroph ic. Definition III.6. Let C b e a noncatastro phic conv olutional code of le ngth n over Z p r . Let E ( z ) ∈ Z κ × n p r [ z ] be a delay- free nonc atastrophic p -encoder for C , such th at the rows of E lr c are p -lin early independ ent in Z n p r . Then E ( z ) is called a minimal p -enco der of C . Furthermo re, the p -indices of C are defined as the r ow degrees of E ( z ) and the p -degree of C is defined as the sum of the p -in dices of C . Thus, in the terminolog y of section II, the rows of a minimal p -encod er ar e a red uced p -basis. If the code C has a canon ical encoder G ( z ) , th en bo th G lr c mod p and G (0) mod p have full r ow rank in Z k × n p , so that a minimal p -enco der is tr ivially constructed as E ( z ) =      G ( z ) pG ( z ) . . . p r − 1 G ( z )      . (4) An important observation is that all noncatastroph ic cod es admit a minimal p -encoder E ( z ) but not all su ch codes admit an encoder G ( z ) that is row reduced and/or delay-free. Definition III.7. Let C be a con volutional co de of len gth n with p - encode r E ( z ) ∈ Z κ × n p r [ z ] . Den ote the sum of th e row degrees of E ( z ) by γ an d let ( A, B , C, D ) ∈ Z γ × γ p r × Z κ × γ p r × Z γ × n p r × Z κ × n p r be a c ontroller canon ical r ealization of E ( z ) . Then th e con- troller ca nonical trellis correspon ding to E ( z ) is defined as X = { X t } t ∈ Z , where X t = ( Z n p r , A γ p , K t ) with K t = { ( s ( t ) , s ( t ) C + u ( t ) D , s ( t ) A + u ( t ) B such that s ( t ) ∈ A γ p , u ( t ) ∈ A κ p } . Note that the states take their values in the nonlinear set A γ p , which is n ot clo sed with respect to add ition or scalar multiplication. Similarly , th e in puts take th eir values in the nonlinear set A κ p . The next theorem presents ou r main result. Theorem III.8. Let C be a no ncatastr o phic co n v olutiona l cod e of length n with minimal p -en coder E ( z ) ∈ Z κ × n p r [ z ] . De note the p -degr ee o f C b y γ . Then the con tr oller canonic al trellis corr espo nding to E ( z ) is a minimal tr ellis repr esentation for C . In particu lar , the minimum n umber of tr ellis states equ als p γ . Pr oof: see Appendix B. In th e field case r = 1 the ab ove th eorem coincid es with the classical result, i.e., the minimu m nu mber of trellis states equals p γ , wh ere γ is the degree of the code. For con volution al codes that admit a canon ical encoder, we have the following c orollary , which follows immediately from applying Th eorem III .8 to the min imal p -enc oder giv en by ( 4). Note that the result coin cides with results in [26 , Sect. 7.4], where a canonical en coder is called “min imal-basic”. Corollary III.9. Let C be a ( n, k ) con v olutiona l code that has a ca nonica l en coder G ( z ) ∈ Z k × n p r [ z ] . Then th e r k p -indices of C are th e k r o w degr ees of G ( z ) , each occurring r times. The m inimum nu mber of tr ellis states equals q ν , whe r e ν is the sum o f the r ow de grees of G ( z ) and wher e q = p r . The next exam ple illustrates o ur theory for the more inter - esting case where the code doe s not admit a canonical encoder . Example III.10 . Over Z 4 : consider the (3 , 2) con volutional code C given by the polynomial encoder G ( z ) =  g 1 ( z ) g 2 ( z )  , where g 1 ( z ) =  z 2 + 1 1 0  and g 2 ( z ) =  2 z 2 1  . Clearly , G ( z ) is a left prime encoder whose contro ller ca non- ical trellis h as 4 3 = 64 trellis states. No te that G lr c does not have full row r ank and th erefore G ( z ) is n ot cano nical. Denote by im G ( z ) th e poly nomial m odule spann ed by the ro ws of G ( z ) . A p - basis for the m odule im G ( z ) is provided b y the rows of the matrix     g 1 ( z ) 2 g 1 ( z ) g 2 ( z ) 2 g 2 ( z )     =     z 2 + 1 1 0 2 z 2 + 2 2 0 2 z 2 1 0 0 2     , which has leading row c oefficient matrix     1 0 0 2 0 0 2 0 0 0 0 2     . The r ow r eduction algorithm o f [ 13, Algorithm 3 .11] is particularly simple in this case: by adding z times the third row to the second row , we obtain the matrix E ( z ) , given by E ( z ) =     z 2 + 1 1 0 2 2 z + 2 z 2 z 2 1 0 0 2     , 6 whose rows are a reduced p -basis f or the module im G ( z ) . Indeed , the ro ws o f its leadin g row co efficient matr ix, gi ven by     1 0 0 0 2 1 2 0 0 0 0 2     , are p -linearly independen t. As a result, the p -in dices o f C are 2 , 1 , 1 , 0 and the p -degree of C equals 4 . The controller canonica l trellis correspond ing to E ( z ) is given by A =     0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0     ; B =     1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0     ; C =     0 0 0 1 0 0 0 2 1 2 0 0     ; D =     1 1 0 2 2 0 0 2 1 0 0 2     . This trellis is minimal with 2 4 = 16 trellis states. Example III.11 . Over Z 4 : consider the (2 , 1) con volutional code C of E xample III.4, giv en by the p olynom ial encoder G ( z ) = [2 2 + z ] (no te that G ( z ) is no t delay- free). The delay-fr ee p -en coder E ( z ) =  2 2 + z 0 2  . of Ex ample I II.4 is clearly minim al, so th at its correspon ding trellis is minimal with 2 states which concurs with [6 ]. I V . C O N C L U S I O N S An impor tant class of po lynomia l encode rs for conv o - lutional co des over a field ar e the cano nical ones. Th eir feedfor ward shift r egister implemen tations are minimal trellis representatio ns o f the code. The trellis state space is linear . Howe ver, for conv o lutional co des over the finite ring Z p r , the literatu re has generalized this result only for restricted cases. In this p aper we intro duce the conce pt of p -en coder and define minimal p - encoder for the c lass of noncatastrophic conv olutio nal codes. W e show how to obtain a min imal p - encoder from a p olyno mial encodin g of the code. W e show that the feedforward shift register implementatio n of such a minimal p -encod er is a minimal tr ellis representation o f the code. Its tre llis state space is non linear . W e also express the minimal nu mber of states in term s of the row d egrees of the minimal p -encod er . I n our vie w a minimal p -encoder is th e ring analogo n of the “canon ical po lynom ial en coder” f rom the field case. W e a lso present the novel con cepts o f p - indices and p -degree of a code as analo gons of the field notion s of “Forney indices” and “degree”, respe ctiv e ly . Our appro ach allows us to view “delay-f reeness” as a prop erty of the p - encode r . Thus we a rrive at the n ovel result that d elay- freeness is a p roperty of the encod ing (just as in the field case) rather than a property of the co de, as in the literature so far (see e.g . [4, su bsect. V -C]). W e conjectu re that a similar phenom enon occu rs with respect to c atastrophicity , i.e., “noncatastro phic” is a property of the p -encoder, not the code. This would imply th at min imal p - encoder s can be o btained fo r all conv olutio nal cod es ov er Z p r , including the catastrophic codes. T his is of particular importance for rotationally in vari- ant catastrophic cod es, see e.g. [1 8]. It is a topic of fu ture research to investigate this conjecture which is likely to in volve a generalization of a typ e of “normal form ” f or polyno mial matrices over Z p r . T o su pport our conjecture, let us examine the rotationally in variant catastrophic code C 1 over Z 4 giv en by the encoder G 1 ( z ) =  3 + 3 z + 3 z 2 3 + z + z 2  . A noncatastro phic minima l p - encoder f or C 1 is gi ven by E 1 ( z ) =  3 + 3 z + 3 z 2 3 + z + z 2 2 2  , yielding a minimal trellis represen tation of C 1 with 4 s tates. Similarly the catastrop hic code C 2 over Z 4 with encod er G 2 ( z ) = [1 + z 1 + 3 z ] has a noncatastrophic minim al p - encoder E 2 ( z ) =  1 + z 1 + 3 z 2 2  , yielding a m inimal trellis r epresentatio n of C 2 with 2 states. V . A C K N O W L E D G M E N T S The auth ors than k the r evie wers f or he lpful comme nts, particularly for alerting us to the rele vance of rotationally in variant c odes. The first autho r is suppo rted in p art by the Au stralian Re- search Council; th e seco nd author is supp orted in par t b y the Portugue se Scienc e Foundation (FCT) throu gh the Unida de de Inv estigac ¸ ˜ ao Matem ´ atica e Aplicac ¸ ˜ oes o f the University of A veiro, Portugal. A P P E N D I X A In this appen dix we recall th e co nstruction of a minimal trellis fo r a con volution al code C as a so-called two-side d r ealizatio n of C , see [25], [6], [21], [15], [16], [26]. Consider two code sequences c ∈ C and ˜ c ∈ C . Con form [25], the concaten ation at time t ∈ Z of c an d ˜ c , denoted by c ∧ t ˜ c , is defined as c ∧ t ˜ c ( t ′ ) :=  c ( t ′ ) for t ′ < t ˜ c ( t ′ ) for t ′ ≥ t . The code sequen ces c and ˜ c are ca lled eq uivalent , denoted by c ≃ ˜ c , if c ∧ 0 ˜ c ∈ C. Definition A.1 . Let C be a line ar co n volutional code of len gth n over a finite r ing R . Th e canonical trellis of C is defined as X = { X t } t ∈ Z , where X t = ( R n , S, K t ) with S := C mod ≃ and K t := { ( s ( t ) , c ( t ) , s ( t + 1)) | s ( t ) = z − t c mod ≃ and s ( t + 1) = z − t − 1 c mod ≃} . It h as b een shown in [25] tha t the above trellis is minimal. Intuitively this is explaine d from the fact that, by constructio n, states cannot be merged . 7 A P P E N D I X B In th is ap pendix we prove Theor em III.8 v ia a bijective mapping fro m the contro ller can onical trellis state set to the trellis state set of the canonical trellis that is d efined in Append ix A . W e first p rovide the p roof for the field case. In our pro of o f The orem III .8, wh ich is the ring case, we are then able to high light the parts that are different from the proof for the field case. Theorem B.1. Let C be a ( n, k ) con volutiona l code of d e g r ee ν over a fin ite field R with canon ical enco der G ( z ) ∈ R k × n [ z ] . Then th e contr o ller canon ical tr ellis corresponding to G ( z ) is a min imal tr ellis r epr esentation for C . In particu lar , the min imum numb er of trellis states equals q ν , wher e q is the size of the fi eld R . Pr oof: Den ote the m emory of C by ν ∗ , i.e., ν ∗ is the maximal Forney index of C . Consider the m apping Θ : R ν 7→ C mod ≃ , giv en by Θ( s ) := [ c ] ≃ , where c ∈ C p asses throug h state s at time 0 . Th e mapping Θ is well-d efined since for any s there exists such a cod e sequence and any two code sequ ences that pass th rough state s at time 0 are obviou sly equiv ale nt. Since the trellis state set C mod ≃ of the can onical trellis of Appen dix A is m inimal, it suffices to p rove that Θ is an isomorph ism, as follows. Surjec ti vity fo llows immediately from the fact that all c ode sequ ences pass throu gh so me state at time 0 . Furthermor e, the map ping Θ is linear since Θ( s 1 + s 2 ) = [ c 1 + c 2 ] ≃ . I t remain s to p rove that Θ is injective. For th is, let s ∈ R ν be such tha t Θ( s ) = 0 . Define u ( − ν ∗ ) ,..., u ( − 2 ) , u ( − 1 ) as elements of R k for which  u ( − ν ∗ ) · · · u ( − 2 ) u ( − 1)       B A ν ∗ − 1 . . . B A B      = s. Define u := ( · · · , 0 , 0 , u ( − ν ∗ ) , · · · , u ( − 2) , u ( − 1 ) , 0 , 0 , · · · ) and let c := G ( z ) u be th e co rrespond ing code sequence. The n clearly c p asses th rough s . From Θ( s ) = 0 it now fo llows that the seq uence c ∧ 0 0 is a code sequence . Denote its state at time 0 by s ′ and its input sequ ence by u ′ . T hen clearly  u ′ ( − ν ∗ ) · · · u ′ ( − 2) u ′ ( − 1)       B A ν ∗ − 1 . . . B A B      = s ′ . W e now prove that s = s ′ , as follows . Firstly , it is clear tha t  c ( − ν ∗ ) · · · c ( − 2 ) c ( − 1)  =  u ( − ν ∗ ) · · · u ( − 2 ) u ( − 1)       D B C B AC · · · 0 D B C · · · 0 0 D · · · . . .      . (5) Furthermo re, f rom the fact that the enco der is delay -free (Property 1 in section I) it fo llows that D = G (0) has full row rank and that u ′ ( ℓ ) = 0 for ℓ < − ν ∗ . As a resu lt, ˆ c ( − ν ∗ ) · · · c ( − 2) c ( − 1) ˜ = ˆ u ′ ( − ν ∗ ) · · · u ′ ( − 2) u ′ ( − 1) ˜ 2 6 6 4 D B C B AC · · · 0 D B C · · · 0 0 D · · · . . . 3 7 7 5 . (6) Since D h as full row rank, the matrix in the above equation also h as full row r ank. Since th e r ight-han d sides o f equa- tions ( 5) and (6) are equal, it then follows that u ( ℓ ) = u ′ ( ℓ ) for − ν ∗ ≤ ℓ ≤ − 1 . As a result s = s ′ . W e now prove that s = 0 . By the above, c ∧ 0 0 is a code sequence that passes thro ugh s at time 0 . Its in put sequence u ′ is of the fo rm ( · · · , 0 , 0 , u ′ ( − ν ∗ ) , · · · , u ′ ( M ) , 0 , 0 , · · · ) , where M ≥ 0 . Here we u sed the fact that the encoder is noncatastro phic (Property 2 in section I). By construction the state of c ∧ 0 0 at time M + ν ∗ + 1 th en eq uals zero . W e now use the row reduce dness of G ( z ) to conclud e th at s = 0 , as follows. Denote the state at time M + ν ∗ by ¯ s . Now r ecall the formu la (3) for the controller cano nical form. Since ¯ sA = 0 , the nonzero components o f ¯ s must be last compon ents in a (1 × ν i ) -block in ¯ s . Also, c ( M + ν ∗ ) = 0 , so that ¯ sC = 0 . By construction , the last rows of the ( ν i × n ) -blocks of C are rows from G lr c and are th erefore linearly ind ependen t. As a result, ¯ s = 0 . Repeating th is argument ag ain and ag ain, we conclud e that u ′ (0) = ... = u ′ ( M ) = 0 an d all states for time ≥ 0 are zero, so that, in par ticular s = 0 , which proves the theorem. Obvio usly , the size of the trellis state set S equals q ν . W e now tu rn to the ring case to pr ove th e analogo n of the above theorem . As com pared to th e field case, the proo f requires some ca re becau se the trellis state set A γ p is not lin ear . Proof of Theorem III.8: Define ν ∗ as th e m aximal p -index of C . Co nsider the mappin g Θ : A γ p 7→ C mod ≃ , gi ven by Θ( s ) := [ c ] ≃ , where c ∈ C passes through s tate s at time 0 . Then Θ can be shown to be well-defined and su rjective, as in the proof of Theorem B.1. Note that Θ is not necessarily a linear m apping . As a result, injecti vity can no longer be proven by sho wing that Θ ( s ) = 0 only for s = 0 , as in the proof of Theorem B.1. Thus, to show that Θ is injective, let s and ˜ s ∈ A γ p be such that Θ( s ) = Θ( ˜ s ) . Let c be the code seq uence that passes throug h s at time 0 , as defined in the proof of Th eorem B.1. Let ˜ c be the analogous co de sequence th at p asses through ˜ s at time 0 . Note that both c an d ˜ c have finite suppo rt. From Θ( s ) = Θ ( ˜ s ) it now follows that the sequ ence c ∧ 0 ˜ c is a co de sequenc e. Denote its state at time 0 b y s ′ and its inpu t sequ ence b y u ′ ∈ ( A κ p ) Z . Since E ( z ) is a delay-free p -e ncoder, the rows of E (0) a re a p -basis (u se also Lemma II.3). By Lem ma 2.8 of [13] ( see also [24]), it no w follows from the fact th at inputs 8 only take values in A p that s = s ′ . The reasoning is as in the proof of T heorem B.1. W e n ow prove that s = ˜ s . By the above, c ∧ 0 ˜ c is a cod e sequence th at passes thro ugh s at time 0 . As in the pro of of Th eorem B.1, it follows that its state equ als zero at time M + ν ∗ + 1 for some M ≥ 0 . Since E ( z ) is a minimal p - encoder, the rows of E lr c are p -linearly independ ent. It n ow follows fro m the fact that states only take values in A p that the state at time M + ν ∗ must also be zero. The reasoning is as in the pro of of Theorem B.1. Repeating this argument again and again, we conclu de th at all states for time ≥ ν ∗ are zero. As a r esult, u ′ (0) = u ′ (1) = · · · = u ′ ( ν ∗ − 1) = 0 , so that s  C AC · · · A ν ∗ − 1 C  = ˜ s  C AC · · · A ν ∗ − 1 C  . 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