A MacWilliams Identity for Convolutional Codes: The General Case

A MacWilliams Iden tit y for Con v oluti onal Co des: The General Case Heide Gluesing-Luerssen ∗ , Gert Sc hneider † Ma y 22, 2008 Abstract: A MacWilli ams Iden tit y for con v olutional co d es will b e established. It makes use of the w eigh t adjacency matrices of the co de and its d ual, based on state sp ace realizations (the control ler canonical form) of the co des in question. The MacWilliams Iden tit y applies to v ario u s notions of dualit y app earing in the lite ratur e on con vol utional co ding th eory . Keyw ords: Conv olutional co d es, control ler canonica l form, we ight distribu tion, weig ht ad- jacency matrix, MacWillia ms identit y MSC (2000): 94B05, 94B1 0, 93B15 1 In tro du ction The w eigh t en umerator of a co de k eeps track of the d istribution of co dew ord we ights and therefore is of paramount imp ortance for the error-correcting capabilities of the co d e in question. On e of the most celebrated results in block co de theory , the MacWilli ams I den tit y Theorem, sta tes that the weigh t enumerator of a block co de completely determines the w eigh t en umerator of th e dual co d e and giv es an explicit transformation form ula. The practical and theoretical implications for blo c k code theory hav e b een studied ev er since, see for instance [15, Ch. 11.3, Ch. 6.5, Ch. 19.2] or [10, Thm. 7.9.5]. F or con vo lutional co d es only partial results conce rn ing a p ossible MacWillia ms Id en tit y could b e established so far. About 30 y ears ago it h as b een sho wn b y a simple example that the classical w eigh t enumerator as int ro duced by Viterbi [21] d o es not ob ey any MacWilli ams t yp e of id en tit y , see [20 ]. In other w ords, this w eigh t en umerator is to o coarse in ord er to yield detailed i nf ormation a b out the du al c o d e. This insigh t ga v e rise to the study of a more r efined weigh t en umerating ob ject, the w eight adj acency matrix (W AM). It has b een in tro d uced in [18], bu t app ears already in different notations earlier in the literature. Ind eed, one can sho w that it basica lly coincides w ith the la b els of the w eigh t en u m erator stat e diagram as considered in [1]. The W AM is defined via a state space d escrip tion of the enco der as in tro d uced in [16]. It is lab eled b y the set of all state pairs ( X , Y ), and eac h en try con tains the w eigh t en um er ator of all outputs asso ciated with the corresp onding state transitions from X to Y . Th e resu lting matrix con tains considerably more information ab out the co d e than the classical w eigh t en umerator men tioned ab o v e. Indeed, it is well- known [18 ], [7] ho w ∗ Universit y of K entucky , Department of Mathematics, 715 Patterson Office T o w er, Lexington, KY 40506- 0027, U SA; heidegl@ms.uky .edu † Universit y of Groningen, D epartment of Mathematics, P . O. Box 407, 9700 AK Groningen, The Neth er- lands; schneider@math.rug.nl 1 to deriv e the latter from th e W AM. Unfortu nately , the matrix b y itself is not an inv ariant of the code, but rather dep ends on the c hoice of the encoder and the state space realization. Ho w ev er, this dep endence can nicely b e describ ed and up on factoring out a suitable group action results in an in v arian t of the co de, the generalized W AM. In a previous article [8] w e studied this in v arian t in detail, and, in particular, we could establish a weak MacW illiams t yp e of identit y for the generalized W AM. It states that a certain transform of any W AM of a giv en co d e results in a matrix having up to ordering the same en tries as an y W AM of the dual co d e. F or the class of co d es with all F orn ey indices b eing at most 1 w e could ev en sh o w th at this ordering is actually induced by a state space isomorphism, whic h can also b e give n explicitly . Of course, the isomorphism d ep ends on the chosen representat ions of th e co de and its d ual. T h is result generalize s a MacW illiams iden tit y established in [1 ] for the class of co des of d egree 1 (that is, on ly one F orney index has the v alue 1 w h ile all other indices are zero). In this pap er we will extend the result to arbitrary C C’s. In other w ords, we w ill establish a MacW illiams Iden tit y for the full class of CC’s. Stated more precisely , given a co de and its dual with chosen state space representa tions we will giv e an explicit tr an s formation of the W AM that will resu lt in the W AM of the du al co d e. The result generalizes the classical MacWilli ams Id entit y for blo ck codes. The main outline of the pap er is as follo ws. In the next section w e will introdu ce the basic notions of con vo lutional co ding theory including stat e sp ace real izations as w ell as t w o blo c k co des closely related to the giv en CC . In Section 3 w e will introd u ce the W AM as well as the MacWilliams transformation matrices, and we will state the MacWilliams Identi ty . Section 4 will b e co mp letely devot ed to th e pro of of the MacWillia ms Identit y and therefore will b e rather tec hnical. A d etailed example will illus tr ate the steps of the MacWilli ams transformation. Finally , in Section 5 we will discuss an al ternative notion of dualit y for CC ’s and translate our result to that notion. The follo wing notation w ill b e used throughout. F or an y d omain R and an y matrix M ∈ R a × b w e denote by im M := { uM | u ∈ R a } a nd k er M := { u ∈ R a | uM = 0 } the image and k ernel, resp ectiv ely , of the canonical linear mapping R a 7− → R b , u 7− → uM asso ciated with M . 2 Preliminaries In t h is section w e will co llect the main notions of con vo lutional co din g theory as needed for this pap er. Let F b e a finite field. A k -dimensional c onvolutional c o de of length n is a submo d ule C of F [ D ] n of the form C = im G := { uG   u ∈ F [ D ] k } where G is a b asic matrix in F [ D ] k × n , i. e., there exists a matrix ˜ G ∈ F [ D ] n × k suc h that G ˜ G = I k . In other w ords, G is noncatastrophic and dela y-free. W e call G an enc o d er and the n umb er δ := max { deg γ | γ is a k -minor of G } is said to b e the de gr e e o f the co de C . A co de having these p arameters is called an ( n, k , δ ) cod e. A basic matrix G ∈ F [ D ] k × n with ro ws g 1 , . . . , g k ∈ F [ D ] n is said to b e minimal if P k i =1 deg( g i ) = δ . F or c haracterizations of minimalit y s ee, e. g., [3, Main Thm .] or [17, Thm. A.2]. It is w ell-kno wn [3, p . 495] that eac h co de C admits a min imal encod er G . The ro w degrees deg g i of a minimal encod er G are uniquely determin ed up to ordering and are calle d the F orney indic es of the co de or of 2 the enco d er. It follo ws that a CC has a constan t enco der matrix if and only if the degree is zero. In that case the co de can b e regarded as a blo c k co de. Throughout the main part of this pap er the dual of a cod e C ⊆ F [ D ] n is defined as b C := { w ∈ F [ D ] n | w v T = 0 for all v ∈ C } . (2.1) In other w ords , the du al co de b C is the orthogonal of C with resp ect to the F [ D ]-bilinear form  ( w 1 , . . . , w n ) , ( v 1 , . . . , v n )  7− → P n i =1 w i v i ∈ F [ D ]. In Section 5 we will address a different n otion of du alit y that h as b een in tro d uced in the literature on CC’s as well, and w e will show ho w our resu lt can b e translate d to that notio n. Dualit y as defin ed in (2.1) has b een considered in, e. g., [1], [2], [4], [17], and [20]. It is w ell kno wn [17, Th m. 7.1 ] that if C is an ( n, k , δ ) co d e, then b C is an ( n, n − k , δ ) cod e. (2.2) F or a blo c k co de V ⊆ F ℓ the dual is d enoted by V ⊥ := { w ∈ F ℓ | w v T = 0 for all v ∈ V } . The differen t n otation b C versus V ⊥ for the dual of a co nv olutional co de C v ersus the du al of a blo c k co d e V will b e helpful later on. The weig ht of conv olutional co d ew ords is d efined straigh tforwardly . F or a p olynomial v ector v = P N j =0 v ( j ) D j ∈ F [ D ] n w e defi n e wt( v ) := P N j =0 wt( v ( j ) ), where wt ( v ( j ) ) is the Hamming w eigh t of the constan t ve ctor v ( j ) ∈ F n . Let C [ W ] ≤ n denote the ve ctor sp ace of p olynomials o ver C in the ind etermin ate W of degree at most n . F or an y sub set S ⊆ F n w e define the weight enu mer ator of S to b e the p olynomial w e( S ) := n X j =0 α j W j ∈ C [ W ] ≤ n , where α j := # { a ∈ S | wt( a ) = j } . Recall t hat the classical MacWilli ams I d en tit y f or blo ck co des states that if C ⊆ F n is a k -dimensional co de and F = F q is a field with q elemen ts, then w e( C ⊥ ) = q − k H  w e( C )  (2.3) with H b eing the MacWilliams transform H : C [ W ] ≤ n − → C [ W ] ≤ n , H ( f )( W ) := (1 + ( q − 1) W ) n f  1 − W 1+( q − 1) W  . (2.4) It should b e ke pt in min d th at H d ep end s on th e parameters n and q . Since throughout this pap er these parameters will b e fixed w e do not indicate them explicitly . A cen tral to ol for the purp ose of our p ap er is the description of a CC by the controll er canonical form (CC F). I t will allo w us to intro d uce the main ob ject, th e W AM, as w ell as t w o blo c k co des asso ciated with a CC th at are crucial for our in vestiga tion. Even though th e CCF can b e found in an y textb o ok on con trol theory , we c ho ose to present it here explicitly since man y of our matrix identi ties later on will rely on the precise form of the matrices. Definition 2.1 Let G ∈ F [ D ] k × n b e a min im al enco der with F orn ey indices δ 1 , . . . , δ r > 0 = δ r +1 = . . . = δ k and degree δ := P k i =1 δ i . Let G hav e the rows g i = P δ i ν =0 g i,ν D ν , i = 1 , . . . , k , where g i,ν ∈ F n . F or i = 1 , . . . , r define the matrices A i = 0 1 . . . 1 0 ! ∈ F δ i × δ i , B i =  1 0 · · · 0  ∈ F δ i , C i =    g i, 1 . . . g i,δ i    ∈ F δ i × n . 3 The c ontr ol ler c anonic al form (CCF) of G is defined as the matrix quadrup le ( A, B , C , E ) ∈ F δ × δ × F k × δ × F δ × n × F k × n where A = A 1 . . . A r ! , B =  ¯ B 0  with ¯ B = B 1 . . . B r ! , C = C 1 . . . C r ! , E = g 1 , 0 . . . g k, 0 ! = G (0) . W e call ( A, B , C, E ) a CCF of the code C ⊆ F [ D ] n if ( A, B , C, E ) is th e CCF of a minimal enco der of C . It is well-kno wn that the C CF describ es the en co ding pro cess of the matrix G in form of a state space system. Ind eed, G ( D ) = B ( D − 1 I − A ) − 1 C + E , see [7, Prop. 2.1, Thm. 2.3]. As a consequence, one has for u = P t ≥ 0 u t D t ∈ F [ D ] k and v = P t ≥ 0 v t D t ∈ F [ D ] n v = uG ⇐ ⇒  x t +1 = x t A + u t B v t = x t C + u t E for all t ≥ 0  where x 0 = 0 . (2.5) W e call F δ the state sp ac e of the enco der G (or of the CC F) and x t ∈ F δ the state at time t . The follo w ing tw o b lo c k co des are naturally associated with a giv en co de. Definition 2.2 F or a co d e C ⊆ F [ D ] n define the asso ciated blo c k codes C const := C ∩ F n and C coeff :=  w ∈ F n   ∃ v = P t ≥ 0 v t D t ∈ C such th at v ˆ t = w for s ome ˆ t ≥ 0  . Ob viously , C const is simp ly the blo c k co d e consisting of th e constan t co dewords in C . Consequent ly , this space is generated by the constan t r ows (if an y) of a min im al enco der matrix G . The co d e C coeff is the space of all constan t v ectors that app ear as co efficien t v ectors of some co dewo rd . It can easily b e describ ed b y usin g a CCF ( A, B , C , E ) f or C . Ind eed, let G = P m t ≥ 0 G t D t , where G t ∈ F k × n . Then obvio usly C coeff = im ( G T 0 , G T 1 , . . . , G T m ) T . Since the co efficien t vecto rs of the ro ws of G are collected in the matrices C an d E , this yields C coeff = im  C E  . (2.6) In [8, Prop. I I .7] it has b een sho wn that the t w o blo c k co des from Definition 2.2 and the corresp ondin g cod es b C coeff and b C const asso ciated with the du al code b C are crosswise m utual duals. Precisely , w e ha ve the follo wing result. Prop osition 2.3 Let C b e an ( n , k , δ ) co de o ve r F q with r p ositiv e F orn ey indices, and let the dual b C ha ve b r p ositiv e F orney indices. Then dim C const = k − r , dim b C const = n − k − b r , an d dim C coeff = k + b r , dim b C coeff = n − k + r . F u r thermore, ( C coeff ) ⊥ = b C const , an d , consequently , q k + ˆ r w e( b C const ) = H  w e( C coeff )  . 3 The W eigh t Adjacency M atrix of a Co de The w eigh t adjacency m atrix as defin ed b elo w has b een introd uced in [18] and studied in detail in [7] as well as [8]. The aim of this sectio n is to presen t the basic prop erties of the w eigh t adjacency matrix for a giv en CC as w ell as to form ulate our main result. Recall from (2.5) that the con troller canonical form of an enco der leads to a state space description of the enco d ing pro cess where th e input is giv en b y the co efficien ts of the message 4 stream while the output is the sequence of co d ew ord co efficien ts. The f ollo w ing matrix collect s for eac h p ossible pair of state s ( X, Y ) the in formation whether via a suitable inpu t u a transition from X to Y is p ossible, i. e., whether Y = X A + uB for some u , and if so, collect s the wei ghts of all asso ciated outputs v = X C + uE . Definition 3.1 Let G ∈ F [ D ] k × n b e a minimal enco der with CC F ( A, B , C, E ). The weight adjac ency matrix (W AM) Λ := Λ( G ) ∈ C [ W ] q δ × q δ of G is d efined to b e the m atrix indexed b y ( X , Y ) ∈ F δ × F δ with the en tries Λ X,Y := we( { X C + uE | u ∈ F k : Y = X A + uB } ) ∈ C [ W ] ≤ n . (3.1) Observe that if δ = 0 the m atrices A, B , C do not exist while E = G . As a consequence, Λ = Λ 0 , 0 = w e( C ) is th e ordinary we ight en um erator of the blo c k co d e C = { uG | u ∈ F k } ⊆ F n . The W AM con tains very detaile d information ab out the co de. The classical path weigh t en umerator[12, p. 154], the extend ed r o w distances [13 ], the activ e bur st d istances [9] as w ell as th e column d istances of the co de can all b e computed from the W AM, see [18], [12, Sec. 3.10] and [7]. F or the relev ance of these distance parameters f or the error-correcting p erforman ce of the co de see [13], [9] Let us illustrate the matrix b y an example. Example 3.2 Let F = F 3 and G =  1 + D 2 2 + D 0 1 0 2  ∈ F [ D ] 2 × 3 and b G =  D + 2 2 + 2 D 2 D + 2  ∈ F [ D ] 1 × 3 . It is easy to see that G and b G are minimal and b asic and satisfy G b G T = 0. T h us , the cod es C := im G and b C := im b G are m utual duals. The CCF’s of the giv en enco d ers G and b G are ( A, B , C, E ) =  0 1 0 0  ,  1 0 0 0  ,  0 1 0 1 0 0  ,  1 2 0 1 0 2  and ( b A, b B , b C , b E ) =  0 1 0 0  ,  1 0  ,  1 0 1 0 2 0  ,  2 2 2   , resp ectiv ely . Using the lexicographic ordering of the states in F 2 (0 , 0) , (0 , 1) , (0 , 2) , (1 , 0) , (1 , 1) , (1 , 2) , (2 , 0) , (2 , 1) , (2 , 2) , (3.2) the asso ciated W AM of G is giv en by Λ =               1 + 2 W 2 0 0 2 W 2 + W 3 0 0 2 W 2 + W 3 0 0 2 W + W 2 0 0 2 W 2 + W 3 0 0 W + 2 W 3 0 0 2 W + W 2 0 0 W + 2 W 3 0 0 2 W 2 + W 3 0 0 0 W + 2 W 3 0 0 2 W + W 2 0 0 2 W 2 + W 3 0 0 2 W 2 + W 3 0 0 2 W + W 2 0 0 W + 2 W 3 0 0 2 W 2 + W 3 0 0 1 + 2 W 2 0 0 2 W 2 + W 3 0 0 0 W + 2 W 3 0 0 2 W 2 + W 3 0 0 2 W + W 2 0 0 2 W 2 + W 3 0 0 2 W 2 + W 3 0 0 1 + 2 W 2 0 0 2 W 2 + W 3 0 0 W + 2 W 3 0 0 2 W + W 2               . 5 F or instance, the en try at p osition (6 , 2) is obtained as follo w s. Since th e 6th state is X = (1 , 2) and the 2nd state is Y = (0 , 1) w e hav e to consider the outputs v = (1 , 2) C + ( u 1 , u 2 ) E = (2 + u 1 + u 2 , 1 + 2 u 1 , 2 u 2 ), wh er e ( u 1 , u 2 ) ∈ F 2 is such that (0 , 1) = (1 , 2) A + ( u 1 , u 2 ) B . S ince (1 , 2) A + ( u 1 , u 2 ) B = ( u 1 , 1) this is th e case if and only if u 1 = 0 and w e see that the ent ry at p osition (6 , 2) is giv en by we { (2 + u 2 , 1 , 2 u 2 ) | u 2 = 0 , 1 , 2 } = w e { (2 , 1 , 0) , (0 , 1 , 2) , (1 , 1 , 1) } = 2 W 2 + W 3 . Lik ewise the W AM asso ciated with the state s pace realizatio n ( b A, b B , b C , b E ) of b C can b e computed as b Λ =               1 0 0 W 3 0 0 W 3 0 0 W 0 0 W 3 0 0 W 2 0 0 W 0 0 W 2 0 0 W 3 0 0 0 W 2 0 0 W 0 0 W 3 0 0 W 3 0 0 W 0 0 W 2 0 0 W 3 0 0 1 0 0 W 3 0 0 0 W 2 0 0 W 3 0 0 W 0 0 W 3 0 0 W 3 0 0 1 0 0 W 3 0 0 W 2 0 0 W               . (3.3) As one can see, the W AM cont ains a considerable amoun t of r edund an cy in its entries. F or further details we refer to [8, Sec. I I I]. It is clear from Definition 3.1 that the W AM dep ends on th e c hosen enco d er G . In order to describ e this dep en dence w e asso ciate with any P ∈ GL δ ( F ) the p ermutatio n matrix P ( P ) ∈ GL q δ ( C ) , w here P ( P ) X,Y = 1 if Y = X P and P ( P ) X,Y = 0 else. (3.4) F urthermore, let Π := {P ( P ) | P ∈ GL δ ( F ) } d enote the group of all su ch p ermutatio n matrices. By defin ition, the matrix P ( P ) corresp onds to the p ermutation on th e set F δ induced b y the isomorphism P . Ob viously , f or any matrix Λ ∈ C [ W ] q δ × q δ and an y P := P ( P ) ∈ Π w e hav e  P Λ P − 1  X,Y = Λ X P ,Y P for all ( X, Y ) ∈ F δ × F δ . (3.5) In [7, Thm. 4.1] it has b een sho wn that for a give n cod e C G 1 , G 2 ∈ F [ D ] k × n are minimal enco d ers of C  = ⇒ Λ( G 1 ) = P Λ( G 2 ) P − 1 for some P ∈ Π . (3.6) As a consequ ence, for a giv en co de C with minimal enco der G and W AM Λ = Λ( G ) the equiv alence class [Λ] := {P Λ P − 1 | P ∈ Π } (3.7) forms an in v ariant of the cod e. It is called the gener alize d W AM of C . F or the rest of this pap er we w ill fi x the follo win g data. General Assumption 3.3 Let F = F q b e a field with q = p s elemen ts. Let C ⊆ F [ D ] n b e an ( n, k , δ ) co de with r n onzero F orney indices and defi n e F := F δ × F δ . F urthermore, let G ∈ F [ D ] k × n b e a minimal enco der of C with the firs t r ro ws corresp onding to the nonzero F orney indices. Let ( A, B , C, E ) b e th e corresp ond ing CCF and Λ b e the asso ciated W AM. Lik ewise, let the du al co d e b C ha ve b r nonzero F orney ind ices and let b G ∈ F [ D ] ( n − k ) × n b e a minimal enco der with the first b r ro ws corresp ondin g to the nonzero F orney ind ices. Let ( b A, b B , b C , b E ) b e the corresp ondin g CCF and denote the asso ciated W AM b y b Λ. 6 Recall from (2.2) that C and b C b oth ha v e degree δ and thus the W AM’s Λ and b Λ are b oth in C [ W ] q δ × q δ . In order to formulat e our main resu lt, we need to in tro du ce a certain transformation ma- trix. Th ey are well known fr om the MacWilliams Identit y for the complete weigh t enumerato r for block co des. Cho ose a primitiv e p -th root of unit y ζ ∈ C ∗ and consider the trace form τ : F q − → F p , a 7− → P s − 1 i =0 a p i . Then we defin e the MacWilli ams matrix to b e H := q − δ 2  ζ τ ( X Y T )  X,Y ∈ F δ ∈ C q δ × q δ . (3.8) No w we are ready to p resen t our main result. Theorem 3.4 Let C and b C and the associated data b e as in General Assumption 3.3. Then there exists some P ∈ GL δ ( F ) suc h that b Λ X,Y = q − k H  ( H Λ T H − 1 ) X P ,Y P  for all ( X, Y ) ∈ F , (3.9) where H is as in (2.4) . As a consequence, the generalized W AM’s satisfy [ b Λ] = q − k H ( H [Λ] T H − 1 ) . (3.10) In other words, the matrix q − k H  H Λ T H − 1  is a repr esen tativ e of the generalize d W AM of b C . Recall that, due to (3.6), the W AM’s for t wo different m inimal enco ders of b C differ by conjugation w ith a suitable matrix P ( P ) ∈ Π. This explains the presence of the matrix P ∈ GL δ ( F ) in (3.9 ). Of course, P dep ends on the chosen enco ders G and b G . In terms of the generalized W AM’s, ho wev er, no sp ecific represen tation of the co de and no transformation matrix are needed an ymore. Note also that in the case w h ere δ = 0, the id en tit y (3.10) immediately leads to the MacWillia ms identit y for blo c k co d es as giv en in (2.3). The pr o of of Theorem 3.4 is rather tec hn ical and will b e presen ted in th e next secti on. The resulting version including an exp licit transformation matrix P w ill b e sum m arized in Theorem 3.4 ′ at the end of the next section. 4 Pro of of Theorem 3.4 Let the data b e as in General Assum ption 3.3. T he follo wing simple prop er ties of the matrices in a CCF will come handy throughout this section. Remark 4.1 The matrices ( A, B , C, E ) as in Definition 2.1 ha ve th e follo win g pr op erties. (i) AB T = 0 , B B T B = B , AA T A = A , (ii) im B T = im ( I r , 0) ⊆ F k and k er B = im (0 , I k − r ) ⊆ F k , (iii) im A ∩ im B = { 0 } , (iv) C const = (k er B ) E := { uE | u ∈ ke r B } and im E = im B T E ⊕ C const , (v) k er A ∩ k er C = { 0 } , (vi) (k er A ) C ∩ C const = { 0 } , The fir st 4 prop er ties are easily verified, see also [8, R em. I I.4, Rem. I I.6]. T h e last t wo prop erties are due to the fact that the enco d er matrix G = B ( D − 1 I − A ) − 1 C + E is minimal. 7 Indeed, n otice that k er A = span F { e j l | l = 1 , . . . , r } , where j l = P l i =1 δ i , and w here e 1 , . . . , e δ denote the standard basis vec tors in F δ . Using G as in Defin ition 2.1 we see that e j l C = g l,δ l , the highest coefficien t v ector of the l th ro w of G . Recalling that for a minimal matrix G the highest co efficien t v ectors g 1 ,δ 1 , . . . , g k ,δ k are linearly ind ep endent and noticing th at C const = span F { g r +1 ,δ r +1 , . . . , g k ,δ k } , one easily deriv es pr op erties (v) and (vi). In the sequel the spaces ∆ := im  I A 0 B  and Ω :=  ( X, Y ) ∈ ∆   X C + Y B T E ∈ C const  (4.1) will pla y a cru cial role. Notice that ∆ = { ( X , Y ) ∈ F | Y = X A + uB for some u ∈ F k } = { ( X , Y ) ∈ F | Λ X,Y 6 = 0 } , (4.2) that is, ∆ is the space of all ordered pairs of states ( X, Y ) admitting a direct transition Y = X A + uB for some suitable input u . Th e set Ω describ es those sta te pairs for wh ic h one of the transitions leads to zero output. Ind eed, w e ha ve Prop osition 4.2 Ω = { ( X, Y ) ∈ ∆ | ∃ u ∈ F k : Y = X A + uB , 0 = X C + uE } . Pr oo f: W e will make use of Remark 4.1(i) and (iv). F or “ ⊆ ” let ( X , Y ) ∈ Ω. Th en Y = X A + uB f or some u ∈ F k and w e compute X C + Y B T E = X C + X AB T E + uB B T E = X C + uB B T E . Since, by assump tion, this v ector is in C const , w e obtain X C + uB B T E = u ′ E for some u ′ ∈ ke r B . No w X C + ( uB B T − u ′ ) E = 0 and therefore ( X, Y ) = ( X, X A + uB ) = ( X, X A + ( uB B T − u ′ ) B ) is in the set on the right hand sid e. F or “ ⊇ ” let Y = X A + uB and 0 = X C + uE . Usin g Remark 4.1(i) and (iv) w e compute X C + ( X A + uB ) B T E = X C + uB B T E = X C + uE + u ( B B T − I ) E = u ( B B T − I ) E ∈ (k er B ) E = C const . As a consequence, ( X, Y ) ∈ Ω . ✷ The follo wing results will b e cru cial. Th e first three s tatemen ts are easily obtained from the f orm of the matrices ( A, B , C, E ) and can b e found in [8, Prop. I I I.6, Lem. I I I.7, Prop. I I I .11]. The last result needs some more d etailed considerations and has b een pr o v en in [8, Lem. I I I.9], where the s p ace Ω app ears as k er Φ. Prop osition 4.3 (a) d im ∆ = δ + r . (b) T he orthogonal of ∆ in F is giv en by ∆ ⊥ = { ( X A T , − X A T A ) | X ∈ F δ } . (c) ∆ ⊕ ∆ − = F , where ∆ − := { (0 , Y ) | Y ∈ im A } . (d) d im Ω = δ − b r , where b r is as in General Assumption 3.3. In the pap er [8], a w eak v ersion of the MacWilliams Iden tit y 3.4 has b een established . In order to present th at result, we define M 0 := b C C T b C E T B b B T b E C T 0 ! ∈ F 2 δ × 2 δ and M 0 := im M 0 . (4.3) 8 Let us also consider the du al versions of the spaces in (4.1) and Pr op osition 4.3(c); that is, let b ∆ , b Ω , and b ∆ − denote the resp ectiv e spaces asso ciated with the d ual co de b C . In the s equel w e will mak e frequent use of th e du al resu lts of Prop osition 4.3. F rom [8, Lem. V.3] it follo ws that b Ω ⊕ b ∆ − = ke r M 0 , M 0 ⊆ Ω ⊥ , and M 0 ⊕ ∆ ⊥ = Ω ⊥ . (4.4) Notice that, according to Prop osition 4.3(a) and (d) and their dual v ersions, dim b Ω = δ − r = dim ∆ ⊥ . No w let us c ho ose a direct complement b ∆ ∗ of b Ω in b ∆ and let G b e a dir ect complemen t of Ω ⊥ in F . Then dim G = 2 δ − ( δ + b r ) = 2 δ − dim b ∆ = dim b ∆ − . F ur thermore, let f 0 : b ∆ ∗ − → M 0 b e the isomorphism ( X, Y ) 7− → ( X , Y ) M 0 . All this leads to th e diagram b ∆ z }| { F f   = b ∆ ∗ f 0   ⊕ b Ω f 1   ⊕ b ∆ − f 2   F = M 0 ⊕ ∆ ⊥ ⊕ G | {z } Ω ⊥ (4.5) where, du e to the d imensions, there exist vecto r space isomorphisms f 1 : b Ω − → ∆ ⊥ and f 2 : b ∆ − − → G in the last t wo columns and where f = f 0 ⊕ f 1 ⊕ f 2 . No w w e can present a cornerstone in the pro of of the MacWilliams Identi ty . T he fol- lo wing w eak ve rs ion of the identit y has b een p ro ve n in [8, T hm. V.5]. I t shows that b Λ and q − k H  H Λ T H − 1 ) ha ve the same entries up to an automorphism f on the sp ace F of state pairs. Theorem 4.4 Cons id er the diagram (4.5) . Th en b Λ f − 1 ( − Y , X ) = q − k H  ( H Λ T H − 1 ) X,Y  for all ( X, Y ) ∈ F . where f is the automorphism on F defined as f := f 0 ⊕ f 1 ⊕ f 2 . It is wo rth b eing stressed that in this theorem the spaces b ∆ ∗ and G are an y arbitrary direct complemen ts of b Ω in b ∆ and of Ω ⊥ in F , resp ectiv ely . Also, the isomorphism s f 1 and f 2 are not f urther sp ecified. In order to pr o v e our main result, Theorem 3.4, we will use this remaining fr eedom su c h that the resulting automorph ism f = f 0 ⊕ f 1 ⊕ f 2 is of the form as desired in Theorem 3.4. More precisely , we n eed f to resp ect the decomp osition F = F δ × F δ , that is, f ( X, Y ) = ( X , Y )  0 P − P 0  for all ( X, Y ) ∈ F (4.6) for some s tate space isomorphism P ∈ GL δ ( F ). Indeed, with f b eing of this form w e obtain f − 1 ( − Y , X ) = ( X P − 1 , Y P − 1 ) and th e identit y in T h eorem 4.4 tur n s in to b Λ X P − 1 ,Y P − 1 = q − k H  ( H Λ T H − 1 ) X,Y  for all ( X , Y ) ∈ F . T h is is exactly the statemen t of Theorem 3.4. T he rest of this section will b e d ev oted to sp ecifying the c hoice of the spaces b ∆ ∗ and G as w ell as the isomorphisms f 1 and f 2 in Diagram (4.5) in order to meet the requirement (4.6) . Let us b egin with the follo wing tec hn ical facts. 9 Prop osition 4.5 (a) Let Π 1 : F − → F δ b e th e pro j ection ont o the fir st comp on ent, thus Π 1 ( X, Y ) = X for all ( X, Y ) ∈ F . Th en Π 1 | Ω is injectiv e. (b) r ank C b E T b B = b r . (c) k er C b E T b B = Π 1 (Ω) = { X ∈ F δ | ∃ u ∈ F k : ( X, X A + uB ) ∈ Ω } . Pr oo f: (a) S u pp ose (0 , uB ) ∈ Ω for some u ∈ F k . Then uB B T E ∈ C const . But then Remark 4.1(i) and (iv) along with the full row rank of E yield uB = uB B T B = 0, w h ic h pro ves (a ). (b) Again we will emplo y Remark 4.1(i) and (iv). Let X ∈ F δ suc h that X C b E T b B = 0. Using (2.6) w e h a v e, on the one hand, X C ∈ C coeff = ( b C const ) ⊥ , where the last iden tit y is due to Prop osition 2.3 . On the other hand, X C ∈ ke r ( b E T b B ) = (im b B T b E ) ⊥ . Making u se of Remark 4.1(iv) and its d ual version this yields X C ∈ ( b C const ) ⊥ ∩ (im b B T b E ) ⊥ = ( b C const ⊕ im b B T b E ) ⊥ = (im b E ) ⊥ . But the latter space is id en tical to im E , as one can see d irectly from the ident ity 0 = G ( D ) b G ( D ) T =  B ( D − 1 I − A ) − 1 C + E  b B ( D − 1 I − b A ) − 1 b C + b E  T and the fu ll ro w rank of the matrices E and b E . Thus we conclude that X C ∈ im E = C const ⊕ im B T E . Usin g that B T B B T = B T , we obtain the existence of some u = ˜ uB T ∈ F k suc h that X C + uB B T E ∈ C const . Along with the identit y AB T = 0 this implies that ( X, X A + uB ) ∈ Ω. All this shows that k er C b E T b B ⊆ Π 1 (Ω) and, using (a), we arrive at d im ke r C b E T b B ≤ dim Π 1 (Ω) = dim Ω = δ − b r . Since C b E T b B ∈ F δ × δ this im p lies b r ≤ rank C b E T b B ≤ r ank b E T b B . Recalling fr om Pr op osition 2.3 that dim b C const = n − k − b r , the dual version of Remark 4.1(iv) along with r ank b E = n − k th en tells us that r an k b E T b B = b r . This finally pro ves rank C b E T b B = b r . (c) The inclusion “ ⊆ ” has b een shown in the pro of of (b). Thus equalit y of the tw o spaces follo w s f rom Prop osition 4.3(d) since dim ker C b E T b B = δ − b r = d im Ω = dim Π 1 (Ω). ✷ P art (a) and (c) of the previous pr op osition giv e rise to a crucial map. Corollary 4.6 Let K := ke r C b E T b B ⊆ F δ . Th en σ : K − → F δ , X 7− → Y suc h that ( X , Y ) ∈ Ω is a we ll-defined , linear, and injectiv e map. F urthermore, K d o es not conta in a nonzero σ -inv ariant subset. Pr oo f: W ell-defin edness follo w s from Prop osition 4.5(a) and (c), w hereas linearit y is ob vious. As for injectivit y , let X ∈ K such that σ ( X ) = 0. Then ( X , 0) ∈ Ω, meaning that X C ∈ C const . On th e other hand, ( X, 0) ∈ Ω ⊆ ∆ tells us that 0 = X A + uB for some u ∈ F k . Hence X A = − uB ∈ im A ∩ im B and Remark 4.1(iii) implies X A = 0. But then X C ∈ (k er A ) C ∩ C const = { 0 } , where the last identit y is due to Remark 4. 1 (vi). As a consequ ence, X ∈ ker A ∩ ke r C , and due to (v) of the same remark w e arriv e at X = 0. This prov es the injectivit y of σ . F or th e last statemen t assume that K ′ is a σ -in v ariant su bset of K . That simply means that there exists some v ector X ∈ K suc h that σ i ( X ) ∈ K for all i ≥ 0. Sin ce K ⊆ F δ is a finite set, th is yields that th e orbit { σ i ( X ) | i ∈ N 0 } is finite and hence con tains a cycle. In other w ords, there exists some X ′ ∈ K and some j > 0 suc h that σ j ( X ′ ) = X ′ . Without loss of generalit y we ma y assum e X ′ = X . By definition of the m ap σ we h a v e  σ i ( X ) , σ i +1 ( X )  ∈ Ω for all i ≥ 0. Using Prop osition 4.2 all this tells us that w e ha ve a cycl e X − − − → ( u 0 0 ) σ ( X ) − − − → ( u 1 0 ) σ 2 ( X ) − − − → ( u 2 0 ) · · · − − − → ( u j − 1 0 ) σ j ( X ) = X 10 of weig ht zero in th e state tr ansition d iagram asso ciated with ( A, B , C, E ). Here the notation X − − − → ( u v ) Y stand s for the equ ations Y = X A + uB , v = X C + uE . It is w ell-kno wn [14, p . 308] that the basicness of the encoder G implies that s uc h a cycle is a concatenatio n of the trivial cycle, that is, X = 0 and u i = 0 for all i = 0 , . . . , j − 1. Th us K ′ = { 0 } and the pro of is complete. ✷ Let us no w introdu ce the matrices S 0 := B T E and S i := B T B A i − 1 C for i ≥ 1 . (4.7) Lik ewise, we d efine the m atrices b S 0 := b B T b E and b S i := b B T b B b A i − 1 b C , i ≥ 1, asso ciated with the dual co de. F urthermore, w e p ut N := X m ≥ 2 m − 1 X i =1 i − 1 X j =0 ( b A T ) i − 1 b S j S T m − j A m − ( i +1) , b N := X m ≥ 2 m − 1 X i =1 i − 1 X j =0 ( A T ) i − 1 S j b S T m − j b A m − ( i +1) . (4.8) Using that A i = 0 = b A i for i ≥ δ it is easy to see that th ese su ms are indeed finite since eac h summand v anish es for m ≥ 2 δ . Using tw o ind ex c hanges one easily shows that b N T = X m ≥ 2 m − 1 X i =1 m X j = i +1 ( b A T ) i − 1 b S j S T m − j A m − ( i +1) . (4.9) In the app endix w e pr o v e the f ollo w ing tec hn ical, but straigh tforward prop erties. Prop osition 4.7 (a) N + b N T = − b C C T . (b) b C S T 0 + b S 0 C T = N A + b A T b N T . (c) N AA T = N . No w w e are in a p osition to w ork on the remaining fr eedom in Diagram (4 .5). Defin e the matrices M 1 =  N − N A 0 0  , M 2 = b N T 0 − b A T b N T 0 ! . (4.10) Recalling that S 0 = B T E , Prop osition 4.7 along with the matrix M 0 defined in (4.3) yields M := M 0 + M 1 + M 2 =  0 P − P 0  , where P := b C S T 0 − N A. (4.11) In the r est of th is section w e w ill show that, fir s tly , th e map f induced by M 0 + M 1 + M 2 is an automorphism, that is, the matrix P ∈ F δ × δ is regular, and, secondly , th at f resp ects the decomp osition of F on th e righ t h and side of Dia gram (4. 5 ). As a co ns equ ence, f d efines an automorphism as in Th eorem 4.4 that, at the same time, is of the form as in (4.6). All this will establish Theorem 3.4. In order to carry out th ese computations n otice th at M 2 = c M T 1 , i. e., M 2 is the du al v ersion of M T 1 . F rom Remark 4.1(i) and Pr op osition 4.7(c) w e obtain  N − N A 0 0   I 0 A T B T  = 0 . 11 Consequent ly , im M 1 ⊆ ∆ ⊥ and b ∆ ⊆ ker M 2 , (4.12) where the second conta inment follo ws fr om the fir st one via dualit y . Th e follo wing result establishes the regularit y of P . Theorem 4.8 Th e matrix P = b C S T 0 − N A is in GL δ ( F ) . Pr oo f: W e need to resort to the du al version of Corollary 4.6. Thus, consid er b K = k er b C E T B with the corresp onding map b σ . Firstly , one observ es that k er P ⊆ b K . In deed, for X ∈ ke r P w e hav e X N A = X b C S T 0 = X b C E T B ∈ im A ∩ im B , and from Remark 4.1(iii) w e conclude X N A = X b C E T B = 0 . (4.13) This prov es k er P ⊆ b K . In ord er to sh o w the regularity of P , let X ∈ ke r P . Then X ∈ b K and th us ( X , b σ ( X )) ∈ b Ω. Recalling that b Ω ⊆ b ∆ w e obtain from (4.12) and (4.4) ( X, b σ ( X )) ∈ k er M 2 ∩ ker M 0 . (4.14) Moreo ver, ( X , b σ ( X )) M 1 = ( X N , − X N A ). But X N A = 0 by (4.13) and thus Pr op osi- tion 4.7(c) y ields X N = X N AA T = 0. Hence ( X, b σ ( X )) ∈ k er M 1 , which along with (4.14 ) implies ( X , b σ ( X )) ∈ k er M . Consequen tly , b σ ( X ) ∈ ker P . All this sh o ws th at k er P is a b σ -inv ariant s ubspace of b K , and b y the d u al v ersion of Corollary 4.6 w e ma y conclude that k er P = { 0 } . Th is yields the desired result. ✷ This theorem sho ws that the map f induced by M = M 0 + M 1 + M 2 is an automorph ism on F of the form as in (4.6). I n order to complete the p ro of of T h eorem 3.4 it only remains to sh o w that f is as in Theorem 4.4, that is, that it resp ects the direct decomp osition as in Diagram (4.5). Th is is accomplished and su m marized in the next result. Prop osition 4.9 Put b ∆ ∗ := ke r M 1 ∩ b ∆ and G := im M 2 . Th en (a) ker M 0 = b Ω ⊕ b ∆ − . (b) ker M 1 = b ∆ ∗ ⊕ b ∆ − . (c) k er M 2 = b ∆ ∗ ⊕ b Ω = b ∆ . (d) im M 1 = ∆ ⊥ and F = Ω ⊥ ⊕ G . Pr oo f: (a) has already b een giv en in (4.4). (b) It is clear from the definition of b ∆ ∗ and the dual version of Pr op osition 4.3(c) that the sum b ∆ ∗ ⊕ b ∆ − is indeed dir ect and con tained in k er M 1 . In order to sh ow equalit y let us first compute the rank of M 1 . T o this end w e show that b Ω ∩ k er M 1 = { 0 } . (4.15) Due to (4.12) and part (a) we hav e that b Ω ⊆ k er M 0 ∩ ker M 2 . Then ( X , Y ) M 1 = ( X , Y ) M for ( X, Y ) ∈ b Ω. No w the r egularit y of the matrix M , see Th eorem 4.8, implies (4.15). Usin g th e dual v ersion of Prop osition 4.3(d) as well as (4.12), w e conclude δ − r = dim b Ω ≤ rank M 1 ≤ dim ∆ ⊥ . Since dim ∆ ⊥ = δ − r du e to Prop osition 4.3(a), this pro ve s rank M 1 = δ − r (4.16) and im M 1 = ∆ ⊥ . (4.17) 12 Next w e show that b ∆ = b ∆ ∗ ⊕ b Ω . (4.18) The directness of the su m on the r igh t hand side as w ell as the inclusion “ ⊇ ” are ob vious, see also (4.15). F ur thermore, notice that k er M 1 + b ∆ = F as b ∆ − ⊆ ke r M 1 and b ∆ − ⊕ b ∆ = F . Since k er M 1 ∩ b ∆ = b ∆ ∗ , w e ob tain with the aid of Prop osition 4.3 that d im b ∆ ∗ = d im(k er M 1 ) + dim b ∆ − dim F = r + b r = dim b ∆ − d im b Ω. All this pro ves (4.18). Along with the dual v ersion of Prop osition 4.3(c) we arrive at F = b ∆ ∗ ⊕ b Ω ⊕ b ∆ − , (4.19) whic h is exactly th e decomp osition of F as in the u pp er r o w of Diagram (4.5). No w w e com- pute dim(k er M 1 ) = δ + r = 2 δ − dim b Ω = dim( b ∆ ∗ ⊕ b ∆ − ), wh ich along w ith b ∆ ∗ ⊕ b ∆ − ⊆ ke r M 1 completes the pro of of (b). (c) Due to (4.18) it only remains to sh o w that b ∆ = ker M 2 . T he in clusion “ ⊆ ” has b een obtained in (4.12). In order to establish identit y recall that M 2 = c M T 1 and therefore dualiz- ing (4.16 ) y ields r an k M 2 = δ − b r . But then dim(k er M 2 ) = δ + b r = dim b ∆. Hence ker M 2 = b ∆, whic h concludes the pr o of of (c). (d) The firs t part has already b een p ro ve n in (4.17) ab o ve . F urtherm ore, from (c) we kno w that dim G = d im(im M 2 ) = δ − b r . Moreo ve r, dim Ω ⊥ = δ + b r . Hence the pr o of of (d) is complete if we can sho w that G ∩ Ω ⊥ = { 0 } . T o th is end assume ( X , Y ) M 2 ∈ Ω ⊥ for some ( X, Y ) ∈ F . B y (c) and (4.19) we ma y assume ( X , Y ) ∈ b ∆ − and ther efore ( X, Y ) M 2 = ( X , Y ) M du e to (a) and (b ). F urthermore, b y (4.4) we hav e Ω ⊥ = im M 0 ⊕ ∆ ⊥ = im M 0 ⊕ im M 1 . As a consequence, the ab o ve yields ( X, Y ) M 2 = ( X 0 , Y 0 ) M 0 + ( X 1 , Y 1 ) M 1 for some ( X i , Y i ) ∈ F , i = 1 , 2. Using (4.19) and (a) and (b) we ma y assume ( X 0 , Y 0 ) ∈ b ∆ ∗ and ( X 1 , Y 1 ) ∈ b Ω. Using once more (a) – (c) we conclude ( X, Y ) M = ( X, Y ) M 2 = ( X 0 , Y 0 ) M 0 + ( X 1 , Y 1 ) M 1 = ( X 0 , Y 0 ) M + ( X 1 , Y 1 ) M and regularit y of the matrix M imp lies ( X, Y ) = ( X 0 , Y 0 ) + ( X 1 , Y 1 ) ∈ b ∆ − ∩ ( b ∆ ∗ ⊕ b Ω) = b ∆ − ∩ b ∆. T hanks to P r op osition 4.3(c) this in tersection is trivial and we may finally conclude that G ∩ Ω ⊥ = { 0 } . This completes the pro of. ✷ The pr op osition shows that the space F decomp oses exactly as in Diagram (4.5) an d that the matric es M i , i = 0 , 1 , 2, indu ce isomorph isms f i , i = 0 , 1 , 2. As outlined in the paragraph righ t after (4.6), Theorem 4.4 along with (4.6), (4.11) and Theorem 4.8 conclude the pro of of Theorem 3.4. W e su mmarize the result as follo w s. Theorem 3.4 ′ Let C and b C and the asso ciated data b e as in General Assumption 3.3. P u t P := b C E T B − N A , where N is as in (4.8) . Th en P ∈ GL δ ( F ) and the W AM’s of C and b C satisfy the MacWilliams Identit y b Λ X,Y = q − k H  ( H Λ T H − 1 ) X P ,Y P  for all ( X, Y ) ∈ F . Consequent ly , the generalized W AM’s [Λ] and [ b Λ] of C and b C satisfy [ b Λ] = q − k H ( H [Λ] T H − 1 ) . W e close this section with illustrating the MacWilli ams Identi ty for the co de in Exam- ple 3.2. Example 4.10 Let the co des C = im G and b C = im b G b e as in Example 3.2. In order to carry ou t the transformation q − k H  H Λ T H − 1  , w e need th e MacWi lliams matrix H . With 13 the same ordering of the states as in (3.2) one obtains H = 1 3               1 1 1 1 1 1 1 1 1 1 ζ ζ 2 1 ζ ζ 2 1 ζ ζ 2 1 ζ 2 ζ 1 ζ 2 ζ 1 ζ 2 ζ 1 1 1 ζ ζ ζ ζ 2 ζ 2 ζ 2 1 ζ ζ 2 ζ ζ 2 1 ζ 2 1 ζ 1 ζ 2 ζ ζ 1 ζ 2 ζ 2 ζ 1 1 1 1 ζ 2 ζ 2 ζ 2 ζ ζ ζ 1 ζ ζ 2 ζ 2 1 ζ ζ ζ 2 1 1 ζ 2 ζ ζ 2 ζ 1 ζ 1 ζ 2               , where ζ = e 2 πi 3 . No w w e may start computing the righ t hand side of the MacWilliams identi ty in Th eorem 3.4’. Using the matrix Λ from Example 3.2 w e obtain H Λ T H − 1 = Γ, wh ere Γ =               f 1 0 0 0 f 4 0 0 0 f 4 0 0 f 4 f 1 0 0 0 f 4 0 0 f 4 0 0 0 f 4 f 1 0 0 0 0 f 3 f 2 0 0 0 f 4 0 0 f 4 0 0 0 f 3 f 2 0 0 f 2 0 0 0 f 4 0 0 0 f 3 0 f 3 0 0 0 f 4 f 2 0 0 f 2 0 0 0 f 3 0 0 0 f 4 0 0 f 4 f 2 0 0 0 f 3 0               with        f 1 = 8 3 W 3 + 4 W 2 + 2 W + 1 3 , f 2 = − 4 3 W 3 + W + 1 3 , f 3 = 2 3 W 3 − W 2 + 1 3 , f 4 = − 1 3 W 3 + W 2 − W + 1 3 . Indeed, using that ζ 2 + ζ + 1 = 0, it can b e c hec ked straightforw ard ly that Γ H = H Λ T . Next one easily compu tes the MacWilliams transf orms 3 − 2 H ( f i ) = W i − 1 for i = 1 , . . . , 4, where H is as in (2.4), and th us one obtains Φ := 3 − 2 H (Γ) =               1 0 0 0 W 3 0 0 0 W 3 0 0 W 3 1 0 0 0 W 3 0 0 W 3 0 0 0 W 3 1 0 0 0 0 W 2 W 0 0 0 W 3 0 0 W 3 0 0 0 W 2 W 0 0 W 0 0 0 W 3 0 0 0 W 2 0 W 2 0 0 0 W 3 W 0 0 W 0 0 0 W 2 0 0 0 W 3 0 0 W 3 W 0 0 0 W 2 0               . Finally , we n eed to apply the state sp ace isomorph ism induced by the m atrix P = b C E T B − N A . Since δ = 2 the matrix N from (4.8) is giv en by N = b S 0 S T 2 + b S 0 S T 3 A + b A T b S 0 S T 3 + b A T b S 1 S T 2 = b S 0 S T 2 + b A T b S 1 S T 2 =  2 0 1 0  . This yields P = b C E T B − N A = „ 1 1 1 2 « . No w one can chec k straigh tforw ardly that Φ X P ,Y P = b Λ X,Y for all ( X, Y ) ∈ F , where b Λ is the W AM of the du al co de giv en in (3.3). T his is exactly the ident it y in Theorem 3.4 ′ . 14 5 Sequence Sp ace Dualit y In this section we w ill b riefly discuss a differen t n otion of dualit y for CC’s and translate the MacWilli ams Id entit y to this t yp e of du alit y . Th u s , thr ou gh ou t this s ection let us call the d ual b C of a co de C ⊆ F [ D ] n as defin ed in (2.1 ) the mo dule-the or etic dual . The literature on conv olutional co ding theory has also seen a notion of dualit y based on the F -b ilinear form F [ D ] n × F [ D ] n − → F ,  X t ≥ 0 v t D t , X t ≥ 0 w t D t  7− → hh X t ≥ 0 v t D t , X t ≥ 0 w t D t ii := X t ≥ 0 v t w T t . Notice th at this s u m is indeed finite since the vect ors are p olynomial. The du al b ased on this bilinear form is usually , and most con v enient ly , defined in the setting of Laurent series, see, e. g., [5], [6], [11], [19]. But we can just as wel l sta y within our p olynomial setting. T hen it amoun ts to defin ing the dual of the code C as e C := { w ∈ F [ D ] n | hh v , D l w ii = 0 for all v ∈ C and l ∈ N 0 } . (5.1) W e call e C the se quenc e sp ac e dual of C . It is easy to see that e C is a submo d ule of F [ D ] n . F urthermore, there is a simple relation b et we en the sequence space dual e C and th e mo dule- theoretic dual b C . Indeed, it is not hard to see th at e C is the time rev ersal of th e mo dule- theoretic dual b C , or, equ iv alen tly , e C is the mo d u le-theoretic dual of the time rev ersal of C , [12, T hm. 2.64]. Here, the time rev ers al co de is obtained f rom the p rimary co d e by rev ersing the time axis. In our purely p olynomial setting, the rev ersal co d e can simply b e defined as follo w s. If G ∈ F [ D ] k × n is a minimal enco der of the co d e C with ro w d egrees δ 1 , . . . , δ k , then it is easy to see that the r e cipr o c al matrix G ′ :=    D δ 1 . . . D δ k    G ( D − 1 ) ∈ F [ D ] k × n (5.2) is m in imal and basic as w ell and h as th e same row degrees δ 1 , . . . , δ k . The co de rev( C ) := im G ′ is called the r eversal c o de of C . Thus, the abov e ma y b e summarized as e C = rev( b C ) = \ rev( C ) . (5.3) W e briefly w ish to mention that in [19 ] y et another n otion of dualit y has b een in tro d uced, based on lo cal br an ch grou p s. I t is lengthy , but straightfo rward to sho w that for conv olutional co des th is t yp e of dualit y is ident ical to sequ ence space dualit y . W e omit the details, but only w an t to p oin t out that the defin ition via local br anc h groups as giv en in [19] has the adv anta ge to circumv ent certain fin iteness issu es arising for sequence space dualit y in the Lauren t ser ies setting. In the r est of th is section w e w ill derive a MacWilliams Ident ity for the d ual pairing ( C , e C ). Due to the close relationship b et wee n the mo d ule-theoretic du al and the sequen ce sp ace dual this can indeed b e dedu ced fr om our p revious r esult. Since a time rev ersal of the state sp ace system in (2.5) essent ially amoun ts to sw apping X and Y in (3.1), it should b e intuitiv ely clear that the W AM of rev( C ) will essen tially b e the transp osed of the W AM of C . Ho w eve r, this is true only when c ho osing the r ight state sp ace repr esentati ons. Th is will b e carried out in the follo wing compu tations. In a first step we need a CCF of rev( C ). 15 Prop osition 5.1 Let the data b e as in Definition 2.1. F urthermore, let G ′ b e the reciprocal matrix of G as in (5.2) . Then the CCF of G ′ is giv en by ( A, B , C ′ , E ′ ) , where  C ′ E ′  =  RA T RB T B R I − B B T   C E  and R =   R 1 . . . R r   ∈ GL δ ( F ) with R i =   1 · · · 1   ∈ GL δ i ( F ) . (5.4) Moreo ver, the matrix L := „ RA T RB T B R I − B B T « satisfies LL T = I δ + k , th us L ∈ GL δ + k ( F ) . Pr oo f: Since G and G ′ are b oth min imal with the same r o w degrees δ 1 , . . . , δ k , it is clear that the CCF of G ′ has, just like G , state transition matrix A and input-to-state matrix B . The iden tities for C ′ and E ′ follo w straightfo rwardly from the form of A, B , C, E as in Definition 2.1 along with the simple matrix identi ties A T A + B T B = I , I − B B T = „ 0 0 0 I k − r « , R = R − 1 = R T , and R A T R = A (5.5) as w ell as the fact that C ′ =      C ′ 1 . . . C ′ r      , wh er e C ′ i =      g i,δ i − 1 . . . g i, 1 g i, 0      , and E ′ =           g 1 ,δ 1 . . . g r,δ r g r +1 , 0 . . . g k , 0           . The iden tit y LL T = I δ + k can easily b e v erified usin g (5.5) and Remark 4.1(i). ✷ No w it is easy to present the W AM of the rev ersal co de. Corollary 5.2 Let Λ b e the W AM of C asso ciated w ith the CC F ( A, B , C, E ) . Th en the W AM Λ ′ of the r ev ersal co d e rev( C ) asso ciated with the CCF ( A, B , C ′ , E ′ ) giv en in Prop o- sition 5.1 satisfies Λ ′ X,Y = Λ Y R,X R for all ( X, Y ) ∈ F , (5.6) where R ∈ GL δ ( F ) is as in (5.4) . Pr oo f: F rom (4.2) w e ha v e Λ ′ X,Y 6 = 0 ⇐ ⇒ ( X , Y ) ∈ ∆ and Λ Y R,X R 6 = 0 ⇐ ⇒ ( Y R, X R ) ∈ ∆. Hence we fi rst ha ve to sho w that ( X , Y ) ∈ ∆ ⇐ ⇒ ( Y R, X R ) ∈ ∆. Using the defin ition of ∆ in (4.1) as w ell as the matrix L from Prop osition 5.1 and the iden tities in (5.5) we obtain im  I A 0 B   0 R R 0  = im  AR R B R 0  = im L − 1  I A 0 B  = im  I A 0 B  . This sho ws that ( X, Y ) ∈ ∆ iff ( Y R, X R ) ∈ ∆ and it remains to p ro ve (5.6) for ( X, Y ) ∈ ∆. F rom [8, Lem. I I I.8] w e kno w that for ( X , Y ) ∈ ∆ Λ ′ X,Y = we( X C ′ + Y B T E ′ + C ′ const ) and Λ Y R,X R = w e  ( Y R ) C + ( X R ) B T E + C const  . (5.7) 16 Since C const is generated b y the constan t ro w s of the minimal enco der G , it follo ws directly from the definition of the recipro cal matrix in (5.2) that C ′ const = C const . F ur th ermore, sin ce ( X, Y ) ∈ ∆, there exists u ∈ F k suc h that Y = X A + uB . Usin g Prop osition 5.1 and (5.5) as w ell as Remark 4.1(i) we compute X C ′ + Y B T E ′ = X R A T C + X R B T E + Y B T B RC + Y B T ( I − B B T ) E = = X ARC + X RB T E + X AB T B RC + uB B T B RC = ( X A + uB ) R C + ( X R ) B T E = ( Y R ) C + ( X R ) B T E With the aid of (5.7) this pro ves (5. 6 ) for all ( X, Y ) ∈ F . ✷ No w it is straightfo rward to formulate an d prov e a MacWilliams Identi ty for the sequence space dual co de. The transformation matrix Q , needed for th e iden tit y , will b e give n explicitly in the pro of. Theorem 5.3 Let C ⊆ F [ D ] n b e as in General Assumption 3.3 and let e C b e the sequence space dual of C . Let ( ˜ A, ˜ B , ˜ C , ˜ E ) b e a CCF of e C and let e Λ b e the associated W AM. Then there exists a matrix Q ∈ GL δ ( F ) suc h that e Λ X Q,Y Q = q − k H ( H Λ H − 1 ) X,Y for all X, Y ∈ F δ . As a consequence, the generalized W AM’s of C and e C satisfy [ e Λ] = q − k H ( H [Λ] H − 1 ) . Pr oo f: Let C ′ = r ev( C ) an d let Λ ′ b e the W AM asso ciated with the CCF ( A, B , C ′ , E ′ ), where C ′ , E ′ are as in Prop osition 5.1. Then e C = b C ′ , due to (5.3). Recall the matrix R from (5.4). By Th eorem 3.4’ we ha ve e Λ X R ˜ P − 1 ,Y R ˜ P − 1 = q − k H ( H Λ ′ T H − 1 ) X R,Y R for all ( X, Y ) ∈ F , where ˜ P := ˜ C E ′ T B − ˜ N A and ˜ N is as in (4.8) with S j and b S j replaced by S ′ j and ˜ S j defined via th e CC F’s ( A, B , C ′ , E ′ ) and ( ˜ A, ˜ B , ˜ C , ˜ E ), resp ectiv ely , as in (4.7). Let P := P ( R ), see (3.4). Usin g (3.5) w e obtain ( H Λ ′ T H − 1 ) X R,Y R =  ( P HP − 1 )( P Λ ′ T P − 1 )( P H P − 1 ) − 1  X,Y . But no w the defin ition of H in (3.8) along with (3.5) and the identit y RR T = I sho ws that P HP − 1 = H . Moreo v er, by C orollary 5.2 and (3.5) w e ha v e P Λ ′ T P − 1 = Λ. All this establishes Theorem 5.3 with the transformation matrix Q := R ˜ P − 1 . ✷ Conclusion W e established a MacWilliams Identit y for conv olutional co des. I t consists of a conjugation of the w eigh t adjacency matrix follo wed by the entrywise MacWillia ms transform ation for blo c k co des. The identi ty applies to b oth m o dule-theoretic dualit y as well as the sequ ence space dualit y . Th e result op ens the do or to in vestiga ting self-du al con volutional co des (with resp ect to an y dualit y notion) with the aid of in v arian t th eory . This will b e pur s ued in a future pro j ect. 17 App endix A In this section w e pro ve th e pu rely m atrix theoretical results of Prop osition 4.7. As b efore, the data are as in General Assump tion 3.3. Since G = E + P i ≥ 1 B A i − 1 C D i w e hav e B T G = P i ≥ 0 S i D i with the matrices S i as giv en in (4.7). Thus, b B T b GG T B = 0 implies m X i =0 b S i S T m − i = 0 for all m ≥ 0 . (A.1) Using the CCF, it is easy to see that P i ≥ 1 ( B A i − 1 ) T ( B A i − 1 ) = I δ . Th is in turn yields C = X i ≥ 1 ( A T ) i − 1 S i (A.2) and, consequen tly , b C C T = X m ≥ 2 m − 1 X i =1 ( b A T ) i − 1 b S i S T m − i A m − i − 1 . (A.3) No w we are ready f or the Pr oo f of Pr opos ition 4.7: (a) F r om (A.1) w e obtain b S i S T m − i = − i − 1 X j =0 b S j S T m − j − m X j = i +1 b S j S T m − j for i = 1 , . . . , m − 1. Using (A.3) and N and b N T from (4.8) and (4.9) w e therefore compute − b C C T = − X m ≥ 2 m − 1 X i =1 ( b A T ) i − 1 b S i S T m − i A m − ( i +1) = X m ≥ 2 m − 1 X i =1 ( b A T ) i − 1  i − 1 X j =0 b S j S T m − j + m X j = i +1 b S j S T m − j  A m − ( i +1) = N + b N T , whic h is wh at w e w ante d . (b) Using again (4.9) one obtains N A + b A T b N T = X m ≥ 2  m − 1 X i =1 i − 1 X j =0 ( b A T ) i − 1 b S j S T m − j A m − i + m − 1 X i =1 m X j = i +1 ( b A T ) i b S j S T m − j A m − ( i +1)  = X m ≥ 2  m − 2 X i =0 i X j =0 ( b A T ) i b S j S T m − j A m − ( i +1) + m − 1 X i =1 m X j = i +1 ( b A T ) i b S j S T m − j A m − ( i +1)  = X m ≥ 2  m − 2 X i =1 ( b A T ) i  m X j =0 b S j S T m − j  A m − ( i +1) + b S 0 S T m A m − 1 + ( b A T ) m − 1 b S m S T 0  Due to (A.1) the inner sum o ver j v an ish es, and adding 0 = b S 0 S T 1 + b S 1 S T 0 , w hic h is (A.1) for m = 1, we pro ceed with N A + b A T b N T = X m ≥ 2  b S 0 S T m A m − 1 + ( b A T ) m − 1 b S m S T 0  + b S 0 S T 1 + b S 1 S T 0 = X m ≥ 1  b S 0 S T m A m − 1 + ( b A T ) m − 1 b S m S T 0  = b S 0 C T + b C S T 0 , 18 where the last iden tit y is a consequence of (A.2). Th is pro v es part (b). (c) As b efore let e 1 , . . . , e δ b e the standard basis v ectors of F δ . T hroughout the rest of this pro of denote, for any m atrix M , the γ -th column (resp. γ -th ro w) of M by M ( γ ) (resp. M ( γ ) ). Let us assume that the m atrices A and B are as in Definition 2.1. T h en w e ha v e k er A = span F { e j l | l = 1 , . . . , r } , where j l = P l a =1 δ a . Moreo v er, AA T is the diagonal matrix with ( AA T ) j l ,j l = 0 for l = 1 , . . . , r and ( AA T ) i,i = 1 else. T herefore, it suffices to sho w that the µ -th column of N is zero for all µ ∈ { j 1 , . . . , j r } . Thus, let µ = P l a =1 δ a for some l = 1 , . . . , r . In order to prov e the desired resu lt w e will ev en s h o w that ( S T m − j A m − i − 1 ) ( µ ) = 0 for all m ≥ 2 and 1 ≤ i ≤ m − 1 as w ell as 0 ≤ j ≤ i − 1. (A.4) This, of course, implies N ( µ ) = 0 due to (4.8). I n order pro v e (A.4 ) notice that S T m − j A m − i − 1 = C T ( A T ) m − j − 1 B T B A m − i − 1 . Put ν := P l − 1 a =1 δ a + 1. The definition of A and B sho ws that ( B A m − i − 1 ) ( µ ) 6 = 0 ⇐ ⇒ ( A m − i − 1 ) ν,µ = 1 ⇐ ⇒ δ l − 1 = m − i − 1 ⇐ ⇒ i = m − δ l . Hence for i 6 = m − δ l w e h av e ( B A m − i − 1 ) ( µ ) = 0 and it remains to p ro ve (A.4) for the case i = m − δ l . In that case 0 ≤ j ≤ m − δ l − 1 implies δ l < m − j . Using that ( B T B ) ( ν ) = e ν , the ν -th ro w of S m − j is ( S m − j ) ( ν ) = ( B T B A m − j − 1 C ) ( ν ) = ( A m − j − 1 C ) ( ν ) = ( A m − j − 1 ) ( ν ) C = 0 , where the last id en tit y follo ws from the simple f act that the l -th diagonal blo c k of A m − j − 1 is zero, as m − j − 1 ≥ δ l . T ransp osing the obtained ident ity yields ( S T m − j ) ( ν ) = 0. Sin ce ( A m − i − 1 ) ( µ ) = ( A δ l − 1 ) ( µ ) = e T ν , w e obtain ( S T m − j A m − i − 1 ) ( µ ) = S T m − j ( A m − i − 1 ) ( µ ) = ( S T m − j ) ( ν ) = 0 . This pro v es (A.4) for the case i = m − δ l and th us conclud es the pro of of Prop osition 4.7. ✷ References [1] K. A. S. Ab d el-Ghaffar. 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