New Construction of 2-Generator Quasi-Twisted Codes

1 Abstract — Quasi-twisted (QT) codes are a generalization o f quasi-cyclic (QC) codes. Based on consta-cyclic simplex codes, a new explicit construction of a family of 2-generator qu asi-twisted (QT) two-we ight codes is presente d. It is also show n that many codes in the fami ly meet the Griesmer bound an d therefore are length-optimal. New distance-opti mal binary QC [195, 8, 96], [210, 8, 104] and [240, 8, 120] co des, and good ternary QC [208, 6, 135] and [221, 6, 144] codes are also obtained by the constr ucti on. Index Terms —linear codes, optimal codes, quasi-cyclic codes, quasi-twisted codes, simp lex codes I. I NTRODUCTION S a generalization to cyclic codes, q uasi-cyclic (QC) codes and quas i-twi sted (QT) c odes have be en shown to contain many good linear codes. Man y researchers have been using modern computers to search for good QC or QT codes, and many record -breaking co des are found [ 1]–[12]. The problem with this method is that it becomes intractable when the dim ensio n and the le ngth of the code becom e larg e. Unfortuna tely , very l ittl e is known on expli cit co nstruct ions of good QC o r QT codes . Fo r 2 -generator QC or QT codes, even fewer results are known [13 ], [14 ]. A linear code is called projective if any two of its coordinates are linearly independent, or in oth er words, if the minimum distance of its dual code is at least three. A code is said to be two-weight if it has o nly two non-zero weights. Projective two-weigh t codes are closely related to str ongly regular graphs [15 ]. In this pap er, a new explicit construction o f a family of 2 - generator QT two -weight codes is pr esented. It is the first time that a family of 2-generator QT cod es is constructed systematically . It is also shown that many codes of this family are good and optimal. Examples are given to show the construction and the modular stru cture of the codes. II. QUASI - TWISTED CODES AN D TWO - WEIGHT CODES A. Consta-Cyclic Codes The code discussed in the following sections is linear. A q-ary linear code is a k-dimensional subspace of an n-dimensional vector space over the fin ite field F q , with minimum distance d between any two codeword s. We denote a q-ary code as an [n, k, d ] q code, or a binary [n, k, d] code if q = 2. A linear [n, k, d] q cod e is s aid to be λ -consta-cyclic if there is a non-zero element λ of F q such that for any Eric Z. Chen is with Dept. of Com puter Science, Kris tianstad University, 291 88 Kristianstad, Sweden( er ic.chen@hkr.se). codeword (a 0 , a 1 , ..., a n-1 ), a consta-cyclic shift by one positi on or ( λ a n-1 , a 0 , ..., a n-2 ) is also a codeword [16 ]. Therefore, the con sta-cyclic code is a generalization of the cyclic code, and a cyclic code is a λ - consta-cyclic code with λ = 1. A consta-cyclic cod e can be defined by a gen erator polynom ial. B. Hamming Co des and Si mplex Cod es Hamming codes are a family of linear single error correcting codes. Fo r any positive integer t > 1 and prime power q, we have a Ham ming [n, n–t, 3] q code, where n = (q t –1)/(q–1). Further, if t and q –1 are relatively prime, then the Hamming code is equivalent to a cyclic code. The dual cod e of a Hamming code is called the simplex code. So fo r any integer t > 1 and prime power q, there is a simp lex [(q t –1)/(q–1), t , q t–1 ] q code. It s hould be note d that a simplex code is an equidistance code, where q t –1 non-zero codewords have a we ight of q t–1. . Let h(x) be a prim itive polynomial of degree t over F q . A λ -consta-cyclic simplex [(q t –1)/(q–1), t , q t–1 ] q code can be d efined by the generator polynom ial g( x) = (x n – λ )/h(x), where n = (q t –1)/(q–1) a nd λ has order of q–1 [16]. Further, a si mplex code is equi valent to a cyclic code if t and q–1 are relatively prime. C. Quasi-Tw isted C odes A code is said to be quasi- twisted (QT) if a consta-cyclic shift of any code word by p p ositions is still a cod eword. Thus a consta-cyclic code is a QT code with p = 1, and a quasi- cyclic (QC) code is a QT code with λ = 1. The length n of a QT code is a mult iple of p, o r n = pm . The consta-cyclic matrices are also called twistulant matrices. They are basic components in the generator matrix for a QT code. An m × m consta-cyclic matrix is defined as , 0 3 2 1 3 0 1 2 2 1 0 1 1 2 1 0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − = c c c c c c c c c c c c c c c c m m m m m m C L M M M M M L L L λ λ λ λ λ λ (1) and the alg ebra of m × m consta-cyclic matrices over F q is isomorphic to the algeb ra in the ring f[x]/(x m – λ ) if C is mapped onto the polynomial formed by the elements of its first row, c(x) = c 0 + c 1 x + … + c m-1 x m-1 , with the least significant coefficient on the left. The polynomial c(x) is also calle d the defi ning poly nomia l of the mat rix C. A twis tulant matrix is called a circulant matrix if λ = 1. The generator matrix of a QT code can be transfo rmed into rows of twis tulant mat rices by s uitabl e perm utation of New Construction of 2-Generator Quasi-Twisted Codes Eric Z. Chen A 2 colum ns. So a 1-genera tor QT c ode has the fol lowing fo rm of the g ener ator matrix [17 ]: G = [ G 0 G 1 G 2 … G p-1 ], (2) where G i, i= 0, 1, …, p–1, are twistulant matrices of order m. A 2-generator QT code has gen erator matrix of the following form: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = 1 , 1 11 10 1 , 0 01 00 ... ... G p p G G G G G G , (3) where G ij are twistulant matrices, for i = 0, 1, and j = 0, 1, …, p–1. Very little on 2-g enerator QT codes is kn own in the literature. Two 2-generato r QC codes were given in [1 3], while a constru ction method for 2-gen erator QC codes was presented in [14]. In both cases, the codes were constructed by the help of modern computers. In the followin g section, we will present an explicit construction o f a family of 2-generator QT codes, that are also two-weight codes, an d we will prove that for many parameters they are good or optimal. D. Two-Weight Codes Let w 1 and w 2 , be the two non-zero weights of a two-weight code, where w 1 ≠ w 2 . We denote a projective q-ar y linear two weight code as an [n, k ; w 1 , w 2 ] q code. There is an on line databa se of two-we ight code s [20]. In the survey pa per [15], Calderbank and Kantor listed many known families of projective two-weigh t codes. Among those families, there is a family of two-weight [n, k; w 1 , w 2 ] q codes, n oted by SU 2. For any prime power q, positive integers t > 1 and i, it has the followi ng param eter s: Block l ength: n = i(q t –1)/(q–1 ) , Dime nsion: k = 2t, Weights : w 1 = ( i – 1) q t-1 , w 2 = iq t-1 , where 2 ≤ i ≤ q t . No expl icit c onstruc tion of t he code i s known and studied in the literature, and very little is known about its structure and performance. III. 2- GENERATOR QU ASI - TWISTED TWO - WEIGHT CODES A. Expli cit Co nstructi on For any integer t > 1 and prime power q, a λ -consta-cyclic simplex [(q t –1)/(q–1) , t, q t-1 ] q can be construct ed. Let g(x) be the generator polyno mial for the λ -consta-cyclic simplex code. Let a 1 , a 2 , …, a q-1 be q – 1 non-zero elements of F q and m=( q t –1)/(q–1). A ny codeword of the λ -consta-cyclic simplex [(q t –1)/(q–1), t, q t-1 ] q can be expressed by a polynomial g i,j (x) = a i x j g(x), with the com putation modul o x m – λ , where i = 1, 2, … , q–1, and j = 0, 1, …, m– 1. Let G t be the twistulant matrix defined by the generator p olynomial g(x), and G i,j be the twistulant matrix defined by g i,j (x). So to tally, we obtain q t –1 twistulant matrices. We construct a generator matrix of a 2-generator QT code as follows: , G G G G G G G G 2 1 1 , 1 1 , 1 ... 0 ... ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = − p t t t (4) where G 1,1 , …, G 1,p–1 are p–1 diff erent twistulant matrices from q t –1 twis tulant mat rices defined a bove, G 1 and G 2 are the first and second ro w of the twistulant matrices of G. Then we have foll owing res ult: Theorem 1: For any p ositive integer t > 1, and prime power q, the generator matrix given in (4) def ines a 2- generator QT two-weig ht [pm, 2t; (p–1)q t-1 , pq t-1 ] q code, where m = (q t –1)/(q–1) and p = 2, 3, … , q t . Proof : Let C 1 be the sub-code defined by G 1 , and C 2 be the sub-code de fined by G 2 . So C 1 consist s of codewords that just repeat the codewords of the consta- cyclic simplex [m, t, q t-1 ] q code by p times. Therefore, C 1 is also an equidistance code with a distance of pq t-1 . Similarly, C 2 is also an equidistance code with a distance o f (p–1)q t-1 , since it s codeword c onsists of first m zeros, f ollowed by (p–1 ) different codewords of the consta-cyclic simplex [m, t, q t-1 ] q code. B ased on the equidistance prop erty of the simplex code and the generator matrix structure of (4 ), the sum of non-zero codewo rds from C 1 and C 2 has a we ight of (p – 1)q t-1 , or pq t-1 , depending on t he codewords i n C 1 and C 2 have the same codeword from the consta-cyclic simplex [m, t, q t-1 ] q code or n ot. Therefore, any non-zero codeword of th e 2-generator QC co de defined b y (4) has a weig ht w 1 = (p–1)q t-1 or w 2 = pq t-1 . This proves Theorem 1. Q.E.D. Theorem 1 provides t he compl ete so lution to t his fam ily of two-wei ght codes . The resul ts prese nted in [21] are two special cases. We state them as corollaries. Wh en q = 2, the binary si mpl ex [2 t –1, t, 2 t-1 ] code is cyclic. So the constructed two-wei ght code i s quasi-cy clic. Coroll ary 2: Let q = 2. For any po sitive integer t > 1, the generator matrix (4) defines a 2-gen erator QC two-weight [p(2 t –1), 2t; (p –1)2 t-1 , p2 t-1 ] code, where p = 2, 3, …, 2 t . If t and q–1 are relatively prime, the sim plex [(q t –1)/(q–1) , t, q t-1 ] q code is also a cyclic code, and thus we have the followi ng result : Coroll ary 3: For any positive integer t > 1, and prime power q. If t and q–1 are relatively prime, the generator matrix (4) defines a 2-genera tor QC t wo-weigh t [pm, 2t; (p–1)q t-1 , pq t-1 ] q code, where m = (q t –1)/(q–1) and p = 2, 3, … , q t . B. Example s Exam ple 1: Let q = 2 and t = 3, we have x 7 –1 = (x + 1) (x 3 + x + 1)(x 3 + x 2 +1). So a binary cyclic simplex [7, 3, 4] code can be defined by g(x) = x 4 + x 2 + x + 1. Let G 3 be the circulant matrix defined b y the generator polynomial g(x), and G 1,j be the circulant matrices defined by g 1,j (x) = x j g(x), where j = 0, 1, 2, …, 6. T hen the following generator matrix defines a binary 2-generator QC two-weight [5 6, 6; 28, 32] code: , G G G G G G G G G G G G G G G G 6 , 1 5 , 1 4 , 1 3 , 1 2 , 1 1 , 1 0 , 1 3 3 3 3 3 3 3 3 0 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = If we take p co lumns of the matrices in the above g enerator matr ix, we o btain ot her bina ry 2-gene rator Q C two-wei ght [14, 6; 4, 8], [21, 6; 8, 12], [28, 6; 12, 16], [ 35, 6; 16, 2 0], [42, 6; 20, 24], a nd [49, 6; 24, 28] codes in the ser ies, for 1< p< 8. Exam ple 2: For q = 3 and t = 3. So m = 13 and q- 1= 2. Since 3 and 2 are relatively prime, we have a cyclic simplex [13, 3, 9] 3 code. It can be s hown that g (x) = x 10 - x 9 + x 8 – x 6 – 3 x 5 + x 4 + x 3 + x 2 + 1 defin es a cy clic s impl ex [13, 3, 9] 3 code. So a serie s of 2-gener ator QC two-weight [13p, 6; 9 (p-1), 9p] 3 can be constructed, with p = 2, 3, ..., 27. Exam ple 3: Let q = 3, and t = 2. So m = 4, q–1 = 2, an d λ =2. Since t and q – 1 are not rel ativel y prim e, we h ave a 2- consta-cyclic [4, 2, 3] 3 code. One pri miti ve polyno mial of degree 2 over F 3 is h(x)=x 2 –x–1. So the c orrespondin g generator polynomial for th is 2-consta-cyclic code is g (x) = (x 4 -2)/h(x )=x 2 +x–1. L et a 1 = 1, a 2 = 2 . Let G 2 be the twist ulant matr ix defi ned by g(x), and G i,j be th e twis tula nt mat rices define d by a i x j g(x), for i = 1 , 2, and j = 0, 1 , 2 and 3. With Theorem 1, a series of 2-generator QT two-w eight [4p, 4; 3(p-1), 3p ] 3 codes can be constructe d for p = 2, 3, ..., 9. IV. GOOD AND OPTIMAL COD ES A. Distance-Optimal Codes A linear [n, k, d] q code is distance-op timal or d-optimal if its minimum distance cannot be improved, i.e., if there does not exist an [n, k, d+1] q code. Fo r given n, k a nd q, ther e is an online table of be st-known l inear c odes [18 ]. It pr ovides t he bound on th e mi nimum dist ance of a code. For s ome parameters, the bounds ar e exact, while for others, the low er and upper bou nds are presented. A code that meets the bound is d-optimal. A code is said to be good if it reaches the lower bound o n the minimum distance, since no codes with larg er distance are know n. For binary quasi-cyclic codes, there is an online database on best-know n binary Q C codes [19]. Am ong the 2-g enerat or QC and QT two-we ight codes construct ed above, m any cod es are good, a nd d-optim al. F or exampl es, t he binary 2-genera tor QC two-wei ght [7p, 6; 4(p- 1), 4p] c odes with 2 < p ≤ 8, and [15p, 8; 8(p-1), 8p] codes with 9 < p ≤ 16 are d-optimal. The 2-gen erator QT two- weight [4p, 4; 3( p-1), 3p] 3 code with 2 < p ≤ 9, the 2-generato r QC [5p, 4; 4(p- 1), 4p] 4 code with 6 < p ≤ 16, and the 2-gener ator QT [6p, 4; 5(p-1), 5p] 5 code wit h 12 < p ≤ 25 are d-optimal too. Among these c odes, [195, 8, 96], [210, 8, 10 4] and [240, 8 , 120] codes are previous ly unknow n to be quas i-cyclic [19]. By computer search for go od codes, Gulliver and Bhargava constructed 1- genera tor QC [36 , 4, 23] 3 code [22]. B ut with q = 3, t = 2, a nd p = 9, a d-opti mal 2-generat or QT [36, 4, 24] 3 code can be constr ucted by our me thod. With q = 3, t = 3, and p = 16 and p = 17, 2-generat or QC two-w eight [2 08, 6; 135, 144] 3 and [221, 6; 144, 153] 3 codes can be o btained and they reach the lower bo und on the di stance [18 ]. B. Length-O ptimal C odes A linea r [n, k, d] q code is length-o ptima l or n-op tim al if its lengt h cannot be impr oved, i.e ., if th ere does not exi st an [n–1, k, d] q code. For an [n, k, d] q code, the bl ock length n is rule d by Grie sme r bound [16]: ∑ − = ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ ≥ 1 0 j q d n k j , (5) where ⎡ x ⎤ denotes t he sm allest intege r greater than or equa l to x. Let p = q t . Then a simp lex code wit h dimens ion 2t ca n be construc ted in a 2-generat or QT form : Theorem 4: For an y prime power q, integer t > 1. Let p = q t and m = ( q t –1)/(q–1). The foll owing gen erator m atri x define s a 2-gener ator QT sim plex [( q 2t –1)/(q–1)= (p +1)m, 2t, q 2t-1 ] q code: . G G G G G G G 0 ... 0 ... 1 , 1 1 , 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = − t p t t t (6) Similar to the proof of Theorem 1, this theor em can be proved. So it is not omitt ed. As the g iven exa mples show tha t ma ny SU2 co des ar e good or distan ce-optim al. Th e follow ing theore m te lls how good the codes are in general. For integers t > 1, j = 1, 2, …, t, and i = 1, 2, …, q t-1 , we defi ne a functi on call ed gap as fo llows: . 1 1) q i ( q) t, gap(i, 1 j ∑ = − = ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ − t j (7) Theorem 5: F or prime power q, i nteger t > 1. Let i, r, and p be intege rs such that i = 1, 2, …, t, r = 1, 2, …, q, a nd p = q t –iq + r +1. T he 2-genera tor QT [p(q t –1)/(q–1), 2t, (p-1)q t-1 ] q code gener ated by the generat or mat rix give n in (4) and (6) me ets t he Griesm er bound wi th a gap give n by gap(i, t , q). Proof: B y the Gri esm er bound, the l ength n of a code of dime nsion k = 2t , and dis tance d = (p-1)q t-1 satisfies , ) 1 ( q d n 1 2 0 1 1 0 ∑ ∑ − = − − = ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ − = ≥ t j j t k j j q q p , ) 1 ( ) 1 ( 1 2 1 1 0 1 ∑ ∑ − = − − = − ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ − + − ≥ t t j j t t j j t q q p q q p n , ) 1 ( ) 1 ... )( 1 ( 1 2 1 ∑ = − − ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ − + + + + − ≥ t j j t t q p q q p n , ) ( ) 1 1 )( 1 ( 1 ∑ = ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ + − + − − − ≥ t j j t t q r iq q q q p n Since ), 1 ( ) 11 1 1 ( − − = + − ∑∑ ∑ == − − = ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ t j t j j j t t j j t q i q q r iq q so we have ), , , ( ) 1 /( ) 1 q ( ) 1 /( ) 1 1)(q (p n q t i gap q q t t − − − + − − − ≥ or, ). , , ( ) 1 /( ) 1 ( q t i gap q q p n t − − − ≥ So the code meets the Griesmer bo und with a gap defined by (7). Q.E.D. If i = 1, gap(1, t, q) = 0. So we have the fo llowing result: Coroll ary 6 : For prime po wer q, integer t > 1. Let r, and p be inte gers such r = 1, 2, … , q, and p = q t –q + r +1 . The [p(q t – 1)/(q – 1), 2t; (p– 1)q t-1 ] q code constructed meets the Griesm er bound wi th equal ity, and thus is l ength-opt imal . As an example, let us consider t = q = 3. The table I lists the 4 calc ulated res ults on the code s obtaine d for p = 17, 18, …, 28. It shows that the code becomes better wh en p increases. In the table , gb denotes the Griesm er bound on the length. Bu t maxi mum p for a 2-ge nerato r QT two- weight co de in t he family is q t , and when p = q t +1, a 2-gen erator Q T simpl ex code is ob tain ed. TABLE I EXAMPLE OF C ODES p d n gb ga p i R q 17 144 221 217 4 4 1 3 18 153 234 230 4 4 2 3 19 162 247 243 4 4 3 3 20 171 260 258 2 3 1 3 21 180 273 271 2 3 2 3 22 189 286 284 2 3 3 3 23 198 299 298 1 2 1 3 24 207 312 311 1 2 2 3 25 216 325 324 1 2 3 3 26 225 338 338 0 1 1 3 27 234 351 351 0 1 2 3 28 243 364 364 0 1 3 3 R EFEREN CES [1] C. L. Chen and W.W. Peterson, “Som e results on quasi-cyclic codes”, Inform. Contr., vol. 15, pp.407–423, 1969. [2] E. J. Weldon, Jr., “Long qua si-cyclic codes are good”, IEEE Trans. Inform. Theory, vol.13, p.130, Jan. 1970. [3] T. Kasami, “ A Gilbert-Varsham ov bound for quasi-cyclic codes of rate 1/2”, IEEE Trans. Inform. Theory, vol. IT-20, p.679, 1974. [4] San Ling and Patrick Solé, “Good self -dual quasi-cyclic code s exist”, IEEE Trans. Inform. Theory, vol.39, pp.1052–1053, 2003. [5] H.C.A. van Tilborg, "On quasi-cyclic codes with rate 1/m", IEEE Trans. Inform. Theory, vol.IT-24, pp.628–629, Sept. 1978. [6] T.A. Gulliver and V.K. Bhargava, "Som e best rate 1/p and rate (p-1)/p systematic quasi-cyclic codes", I EEE Trans. Inform. Theory, vol.IT-37, pp.552–555, May 1991. [7] E. Z. Chen, "Six new binary quasi -cyclic codes", IEEE Tr ans. Inform. Theory, vol.IT-40, pp.1666–1667, Sept. 1994. [8] R. N. Daskalov, T. A. Gulliver a nd E. Metodieva, “New good quasi- cyclic ternary and quaternary linear codes”, IEEE Trans. Inform. T heory, vol. 43, pp. 1647–1650, 1 997. [9] P. Heijnen, H. C. A. van Tilborg, T. Verhoeff, and S. We ijs, "Some new binary quasi-cyclic codes ", IEEE Trans. Inform. Theory, vol. 44, pp. 1994–1996, Sept. 1998. [10] N. Aydin, I. Siap, and D. Ray-Chaudhury, “The structure of 1-generator quasi-twisted codes and new linear codes”, Design, Code s, and Cryptography, 24, pp. 313–326, 2001. [11] R. Daskalov and P. Hristov, “New quasi-twisted degenerate ternar y linear codes”, IEEE Trans. Inform. Theory, vol. 49, pp. 2259–2263, 2003. [12] R. Daskalov, P. Hristov and E. Me tod ie v a, ”N ew mimi mu m di st a nc e bounds for linear codes over GF(5)”, Discrete Math., vol. 275, pp. 97– 110, 2004. [13] T. A. Gulliver and V. K. Bhargava , “Two new rate 2/p binary quasi- cyclic codes”, IEEE Trans. Inform. Theory, Vol. 40, pp. 1667–1668, Sept. 1994. [14] E. Z. Ch en, “New quasi-cyclic c odes from simplex codes”, IEEE Trans. Inform. Theory, vol. 53, pp. 1193–1196, March 2007. [15] R. Calderbank and W. M. Kantor, “The geom etry of two-weight codes” , Bull. London Math. Soc., vol. 18, pp. 97–122, 1986. [16] E. R. Berlekamp, Algebraic Codi ng Theory, Revised 1984 Edition, Aegean Park Press, 1984. [17] G. E. Séguin and G. Drolet, “The theory of 1-generator qua si-cyclic codes”, manuscript, Dept of Elect r. and Comp. Eng., Royal Military College of Canada, Kingston, Ontario, June 1990. [18] M. Grassl , Bounds on the minim um distance of linear codes, [Online]. Available: http://www.codetables.de [19] E. Z. Chen, Web database of binary QC codes, [Online], http://moodle.tec.hkr.s e/~chen/research/c odes/searchqc2.htm [20] E. Z. Chen, Web database of two-weight codes, [online], http://moodle.tec.hkr.se/~chen/res earch/2-weight-c odes/search.php [21] E. Z. Chen, “New constructions of a family of 2-generator quasi-cyclic two-weight codes and related codes”, Proc. of 2007 IEEE Internat. Symp. on Inform . Theory (ISIT2007), pp. 2191–2195, Nice, France, 2007. [22] T. A. Gulliver an d V. K. Bhargava, “S om e best rate 1/p and rate (p-1)/p systematic quasi-cyclic codes over GF(3) and GF(4)”, IEEE Trans. Inform. Theory, vol. 38, pp. 1369–1374