New Linear Codes from Matrix-Product Codes with Polynomial Units

New Linear Co des from Matrix-Pro duct Co des with P olynomia l Units Fernando Hernando ∗ Departmen t of Mathematics, Universit y College Cork Cork, Ireland Diego R uano † Departmen t of Mathematical Sciences, Aalb org University 9220-Aalborg Øst, Denmark Abstract A new construction of codes from old ones is considered, it is an extension of the ma trix -pro duct construction. Several linear codes that impro ve the parameters of the known ones are presented. 1 In tro d uction Matrix-Pr o duct co des were initially considered in [1 , 8]. They are an extensio n of s everal classic co nstructions o f c o des fro m old ones, like the Plotkin u | u + v - construction. In this article we consider this construction with cyclic co des, matr ix- pro duct co des with po ly nomials units, wher e the element s o f the matrix used to define the codes are polyno mials instead of elements of the finite field. The co des obtained with this construction are quasi-cyclic co des [7]. These co des b eca me impo rtant after it was shown that some co des in this class meet a mo dified Gilber t- V arshamov bound [6]. An extension of the lower bound o n the minimum dista nce from [8] is obtained. This b ound is sharp for a matrix-pro duct code o f nested co des, ho wever it is not sharp in this new setting, that is w e obtain code s with minim um distance b eyond this b ound. By investigating the construction of the words with pos sible minimum weigh t o f a matrix- pro duct code, we are able to sift an exha us tive search and to obtain three matr ix-pro duct codes w ith p olynomials units, that improve the pa- rameters of the codes in [4]. Another fo ur linear co des, improving the parameters of the known linear co des, are o btained fro m the previous ones. ∗ Is supported in part by the Claude Shan non Institut e, Science F oundat ion Ireland Grant 06/MI/006 (Ireland) and by MEC MTM2007-64704 and Junta de CyL V A025A07 (Spain). † Is supp orted in part by MEC MTM2007-64704 and Junta de CyL V A065A07 (Spain). 1 2 Matrix-Pro d uct Co des with Pol yn omial Units Let F q be the finite field with q elements, C 1 , . . . , C s ⊂ F m q cyclic co des of le ng th m and A = ( a i,j ) an s × l -matrix, with s ≤ l , whos e en tries a re units in F q [ x ] / ( x m − 1). A unit in F q [ x ] / ( x m − 1) is a p olynomial of degree lower than m who se greatest common diviso r with x m − 1 is 1 (they are co -prime). W e remark, tha t the cyclic co des genera ted by f a nd by f u , with f | x m − 1 and gcd( u , x m − 1) = 1, are the same co de. The so-c alled matrix- pr o duct co de with polyno mial units C = [ C 1 · · · C s ] · A is the set of all matrix- pro ducts [ c 1 · · · c s ] · A where c i ∈ C i ⊂ F q [ x ] / ( x m − 1 ) for i = 1 , . . . , s . The i -th column of any co deword is an element of the form P s j =1 a j,i c j ∈ F q [ x ] / ( x m − 1), the co dewords can be viewed as, c =   s X j =1 a j, 1 c j , . . . , s X j =1 a j,l c j   ∈ ( F q [ x ] / ( x m − 1)) l . One can g enerate C with the matrix : G =      a 1 , 1 f 1 a 1 , 2 f 1 · · · a 1 ,s f 1 · · · a 1 ,l f 1 a 2 , 1 f 2 a 2 , 2 f 2 · · · a 2 ,s f 2 · · · a 2 ,l f 2 . . . . . . · · · . . . · · · . . . a s, 1 f s a s, 2 f s · · · a s,s f s · · · a s,l f s      , where f i is the generator p olynomia l of C i , i = 1 , . . . , s . That is , we ha ve that C = { ( h 1 , . . . , h s ) G | h i ∈ F q [ x ] with degr e e < m − deg ( f i ) , i = 1 , . . . , s } and it follows that C is a quas i- cyclic co de. Prop ositi on 1. L et C i b e a [ m, k i , d i ] cyclic c o de, then the mat rix-pr o duct c o de with p olynomial units C = [ C 1 · · · C s ] · A is a line ar c o de over F q with length l m and dimension k = k 1 + · · · + k s if the matr ix A has ful l ra nk over F q [ x ] / ( x m − 1) . Pr o of. The length follo ws from the construction of the co de. Let A be a s × l matrix with s ≤ l which has full ra nk. Let c i ∈ C i for i = 1 , . . . , s such that [ c 1 , . . . , c s ] 6 = [0 , . . . , 0]. Since A has ra nk e q ual to s then [ c 1 , . . . , c s ] · A 6 = [0 , . . . , 0]. Therefore, # C = # { [ c 1 , . . . , c s ] · A | c i ∈ C i , i = 1 , . . . , s } = (# C 1 ) · · · (# C s ) = q k 1 + ··· + k s . W e denote by R i = ( a i, 1 , . . . , a i,l ) the element o f ( F q [ x ] / ( x m − 1)) l consisting of the i -th row o f A , for i = 1 , . . . , s . W e consider C R i , the F q [ x ] / ( x m − 1)- s ubmo dule of ( F q [ x ] / ( x m − 1)) l generated by R 1 , . . . , R i . In other w or ds, C R i is a linea r co de ov er a ring, and w e denote b y D i the minimum Hamming weigh t of the w ords of C R i , D i = min { w t ( x ) | x ∈ C R i } . W e obtain a lower bound for the minim um distance o f C by just extending the pro of o f the main result in [8]. Prop ositi on 2. L et C b e the matrix-pr o duct c o de with p olynomial u nits [ C 1 · · · C s ] · A wher e A has ful l r ank over F q [ x ] / ( x m − 1) . Then d ( C ) ≥ d ∗ = min { d 1 D 1 , d 2 D 2 , . . . , d s D s } , (1) wher e d i = d ( C i ) , D i = d ( C R i ) and C R i is as describ e d ab ove. 2 Pr o of. An y co deword o f C is of the form c = [ c 1 · · · c s ] · A . Let us suppose that c r 6 = 0 a nd c i = 0 fo r all i > r . It follows that [ c j, 1 x j − 1 , · · · , c j,s x j − 1 ] · A ∈ C R r for j = 1 , . . . , m , where c i = c 1 ,i + c 2 ,i x + · · · + c m,i x m − 1 . Since c r 6 = 0 it has at least d r monomials with no n-zero c o efficien t. Supp ose c i v ,r 6 = 0, fo r v = 1 , . . . , d r . F o r each v = 1 , . . . , d r , the product [ c i v , 1 x j − 1 , · · · , c i v ,s x j − 1 ] · A is a non-zer o co deword in C R r , since A has full rank. Ther efore the w eig ht of [ c i v , 1 x i v − 1 , · · · , c i v ,s x i v − 1 ] · A is greater than or equal to D r and the weight of c is greater than or equa l to d r D r . Remark 1. If C 1 , . . . , C s ⊂ F m q are linear co des of length m and A = ( a i,j ) ∈ M ( F q , s × l ) a matr ix with s ≤ l , then C = [ C 1 · · · C s ] · A is a matrix- pr o duct co de, initially considered in [1, 8]. W e deno te b y R i = ( a i, 1 , . . . , a i,l ) the element of F l q consisting of the i -th row of A , for i = 1 , . . . , s . W e set D i the minimum distance of the co de C R i generated by h R 1 , . . . , R i i in F l q . In [8] the following low er b ound for the minimum distance of the matrix-pro duct co de C is obtained, d ( C ) ≥ min { d 1 D 1 , d 2 D 2 , . . . , d s D s } , where d i is the minimum distance of C i . If we consider C 1 , . . . , C s nested co des, the previo us b o und is sharp for matrix -pro duct co des [5]. Ho wev er , if we c onsider a matrix-pro duct co de with p olynomia l units, then the bo und from prop o sition 2 is not sharp in genera l, a s o ne c an s ee in the examples stated b elow. Let us consider the same a ppr oach a s that of [5 ] to construct a co deword with minim um weight in this mor e general setting: s et c 1 , . . . , c p ∈ F q [ x ] / ( x m − 1) such that c 1 = · · · = c p , with w t ( c p ) = d p , and c p +1 = . . . = c s = 0. Let r = P p i =1 r i R i , with r i ∈ F q [ x ] / ( x m − 1), b e a w o rd in C R p with weight D p . If c ′ i = r i c i then [ c ′ 1 · · · c ′ s ] · A = c 1   p X j =1 a j, 1 r j , . . . , p X j =1 a j,l r j   = c p r . Although, for a cyclic co de C and a unit g in F q [ x ] / ( x m − 1), C = { cg | c ∈ C } , the w eight o f c is different fr o m the o ne of cg , in general. Hence, the weigh t of c p r is g reater than or eq ual to d p D p . W e r emark tha t this phenomenon allows us to obtain co des with minimum distance b e yond the low er bo und. 3 New linear co des: Plotkin construction with p oly- nomials Obtaining a sharp er b ound than the one in the pr evious section is a very toug h problem, actua lly it is the s ame question as the computation of the minimum dis- tance o f a quasi- cyclic co de. How ever, b y analyzing the low er bound d ∗ we hav e per formed a search to find co des with go o d para meters. An exhaustive sear ch in this family is o nly feasible if one c o nsiders some extra co nditions, these conditions should b e necessary for having go od pa rameters, but no t sufficient. W e will as - sume further particular conditions that allow ed us to s uccessfully ach ieve a sea rch, discarding a significan t amoun t of cases. W e hav e used the s tr ucture obtained in the previous sectio n for matr ix-pro duct c o des with p oly nomials units from nested co des and we have o btained some bina ry linear co des improving the pa rameters of the previously known co des . 3 Let s = l = 2, and A the matrix A =  g 1 g 2 0 g 4  , where g 1 , g 2 , g 4 are units in F 2 [ x ] / ( x m − 1). In this wa y A is full rank over F 2 [ x ] / ( x m − 1) with D 1 = 2 a nd D 2 = 1 . W e may also consider this family of co des as an ex- tension o f the Plo tkin u | u + v - construction. F or nested matrix- pro duct co de s the b ound d ∗ = min { d 1 D 1 , . . . , d s D s } is sharp. F urthermore, by [5, Theorem 1] we hav e some words with weigh t d i D i for i = 1 , . . . , s . W e follow the construction of these words a nd consider a matrix A in a such a w ay that they have weigh t la rger than d i D i . Let C 1 = ( f 1 ) and C 2 = ( f 2 ), with f 1 | f 2 (that is, C 1 ⊃ C 2 ), and C = [ C 1 C 2 ] · A . W e co nsider h 1 , h 2 ∈ F 2 [ x ] such that wt ( f 1 h 1 ) = d 1 and wt ( f 2 h 2 ) = d 2 , and r 1 , r 2 ∈ F 2 [ x ] / ( x m − 1) suc h that r 1 R 1 + r 2 R 2 is a co deword with minim um Hamming weigh t in C R 2 , that is with weigh t 1. Thus, the words [ f 1 h 1 , 0] · A = ( f 1 h 1 g 1 , f 1 h 1 g 2 ) and [ f 2 h 2 r 1 , f 2 h 2 r 2 ] · A = ( f 2 h 2 r 1 g 1 , f 2 h 2 ( r 1 g 2 + r 2 g 4 )) hav e weigh t g reater than or equal to 2 d 1 and d 2 , resp ectively . In par ticular, the words with minimum Hamming w eig ht in C R 2 are genera ted by R 2 , for r 1 = 0, and g 4 R 1 − g 2 R 2 , for r 1 = g 4 , r 2 = − g 2 . Therefore, the words o f C with p o ssible minimum weigh t are: ( f 1 h 1 g 1 , f 1 h 1 g 2 ), (0 , f 2 h 2 g 4 ) and ( f 2 h 2 g 1 g 4 , 0). Hence, we wan t to get f 1 h 1 g 1 or f 1 h 1 g 2 with w eig ht greater than d 1 and f 2 h 2 g 4 and f 2 h 2 g 1 g 4 with weigh t g reater than d 2 . W e shall ass ume a ls o that d 2 > 2 d 1 , therefore w e only should hav e f 1 h 1 g 1 or f 1 h 1 g 2 with weight greater tha n d 1 in order to hav e a chance to improv e the lower bo und from Pr op osition 2. Moreov e r we may consider g 1 = 1 without fur ther restr ic tio n o f genera lity: notice that f 2 and f 2 g 1 define the same cyclic co de, hence a co dew o r d is of the form ( f 1 h 1 g 1 , f 1 h 1 g 2 + f 2 h 2 g 1 ). Multiplying by g − 1 1 we obtain ( f 1 h 1 , f 1 h 1 ( g 2 /g 1 ) + f 2 h 2 ) where g = g 2 /g 1 is a unit. Summarizing, we hav e p erformed a sifted search following the criteria : we con- sider matrix-pro duct co des with p o lynomial units C = [ C 1 C 2 ] · A , where C 1 , C 2 are cyclic nes ted code s , with same length and d 2 larger than 2 d 1 , and a matrix A =  1 g 0 1  , with g unit in F 2 [ x ] / ( x m − 1) such that w t ( f 1 h 1 g ) > d 1 . W e hav e compar e d the minim um distance of thes e binary linear co des with the ones in [4] using [2]. W e pre-co mputed a table containing all the cyclic co des up to length 55, their parameters a nd their words of minimum w eig ht. W e o btained the following linea r co des whose pa rameters are b etter than the o ne s previously known: F rom [4] New codes [94 , 25 , 2 6] C 1 = [94 , 25 , 27] [102 , 28 , 27] C 2 = [102 , 2 8 , 28] [102 , 29 , 26] C 3 = [102 , 2 9 , 28] C 1 = [ C 1 , C 2 ] · A , where C 1 = ( f 1 ) a nd C 2 = ( f 2 ) w ith: • f 1 = x 23 + x 22 + x 21 + x 20 + x 18 + x 17 + x 16 + x 14 + x 13 + x 11 + x 10 + x 9 + x 5 + x 4 + 1 , 4 • f 2 = ( x 47 − 1) / ( x + 1) , • g = x 20 + x 19 + x 13 + x 12 + x 11 + x 9 + x 7 + x 4 + x 3 + x 2 + 1 . C 2 = [ C 1 , C 2 ] · A , where C 1 = ( f 1 ) a nd C 2 = ( f 2 ) w ith: • f 1 = x 25 + x 23 + x 22 + x 21 + x 20 + x 18 + x 16 + x 11 + x 10 + x 8 + x 7 + x 6 + x 5 + x 4 + x +1 , • f 2 = ( x 51 − 1) / ( x 2 + x + 1 ) , • g = x 20 + x 15 + x 14 + x 10 + x 9 + x 7 + 1 . C 3 = [ C 1 , C 2 ] · A , where C 1 = ( f 1 ) a nd C 2 = ( f 2 ) w ith: • f 1 = x 24 + x 23 + x 21 + x 19 + x 18 + x 15 + x 14 + x 13 + x 12 + x 11 + x 9 + x 8 + x 6 + x 4 + 1 , • f 2 = ( x 51 − 1) / ( x 2 + x + 1 ) , • g = x 50 + x 49 + x 48 + x 46 + x 44 + x 43 + x 42 + x 41 + x 38 + x 37 + x 36 + x 34 + x 32 + x 29 + x 27 + x 25 + x 24 + x 19 + x 17 + x 15 + x 13 + x 12 + x 10 + x 8 + x 5 + x + 1 . Moreov e r o p e rating on C 3 we g et four mor e co des. F rom [4] New codes Metho d [101 , 29 , 26] C 4 = [101 , 2 9 , 27] Puncture Co de( C 3 ,102) [101 , 28 , 26] C 5 = [101 , 2 8 , 28] Shorten Code( C 3 ,101) [100 , 28 , 26] C 6 = [100 , 2 8 , 27] Puncture Co de( C 5 ,101) [103 , 29 , 27] C 7 = [103 , 2 9 , 28] Extend Code( C 3 ) Also, a go o d num ber of new quasi-cy clic co des re a ching the b est kno wn low er bo unds a r e a chiev ed with this method. O ne can find 43 4 of these co des in [3]. Ac kno wledgemen ts The authors would like to thank M. 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