ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.784999 J. Algebra Comb. Discrete Appl. 7(3) • 237–245 Received: 23 May 2019 Accepted: 25 March 2020 Journal of Algebra Combinatorics Discrete Structures and Applications Generating generalized necklaces and new quasi-cyclic codes∗ Research Article Rumen Daskalov, Elena Metodieva Abstract: In many cases there is a need of exhaustive lists of combinatorial objects of a given type. We consider generation of all inequivalent polynomials from which defining polynomials for constructing quasi- cyclic (QC) codes are to be chosen. Using these defining polynomials we construct 34 new good QC codes over GF(11) and 36 such codes over GF(13). 2010 MSC: 94B05, 94B65 Keywords: Finite field, Quasi-cyclic linear codes, Necklaces 1. Introduction Let GF(q) denote the Galois field of q elements and let V(n,q) denote the vector space of all ordered n-tuples over GF(q). The Hamming weight of a vector x, denoted by wt(x), is the number of nonzero entries in x. A linear code C of length n and dimension k over GF(q) is a k-dimensional subspace of V(n,q). Such a code is called [n,k,d]q code if its minimum Hamming distance is d. For linear codes, the minimum distance is equal to the smallest of the weights of the nonzero codewords. A k ×n matrix G having as rows the vectors of a basis of a linear code C is called a generator matrix for C. Let Ai denote the number of codewords of C with weight i. The weight distribution of C is the list of numbers Ai. The weight distribution A0 = 1, Ad = α, . . . , An = γ is expressed as 01dα . . .nγ also. In order to obtain a q-ary linear code which is capable of correcting most errors for given values of n, k, and q, it is sufficient to obtain an [n,k,d]q code C with maximum minimum distance d among all such codes or for given values of k, d, and q, to obtain an [n,k,d]q code C whose length n is the smallest one. The respective codes in these two cases are called optimal. ∗ This work was partially supported by the Bulgarian Ministry of Education and Science under Contract in TU– Gabrovo. Rumen Daskalov (Corresponding Author), Elena Metodieva; Department of Mathematics and Informatics, Tech- nical University of Gabrovo, Bulgaria (email: daskalov@tugab.bg, metodieva@tugab.bg). 237 https://orcid.org/0000-0001-7441-4757 https://orcid.org/0000-0001-5360-4762 R. Daskalov, E. Metodieva / J. Algebra Comb. Discrete Appl. 7(3) (2020) 237–245 The problem of determining optimal code parameters, known as the Main Linear Coding Theory Problem, has two aspects. One is the construction of codes which optimize minimum distance and the other is proving non-existence of codes of certain parameters ([14], [21]). In the former one often uses computers, but this approach becomes ineffective when the dimension of the codes is large, because, as we know, computing the minimum distance of linear codes is an NP-hard problem [30]. Thus, it becomes expedient to use classes of codes that have a rich mathematical structure. In recent years it has been shown that QC and QT codes form such nice classes. Being generalizations of cyclic and consta-cyclic codes, they contain many good, record-breaking codes [1–6, 10, 11, 15–20, 24, 27]. Markus Grassl and Eric Chen maintain online tables of linear codes. The Grassl’s tables [28] contain lower and upper bounds on minimum distances for linear codes over small finite fields (q ≤ 9). Many of the best-known codes in these tables are QC and QT codes. The Chen’s table [9] contains only good and best-known QC and QT codes (q ≤ 13). These two databases are updated when new codes are discovered. In recent years there has been an increased interest in codes over GF(11) and GF(13). In [25] and [26] Gulliver constructed QT codes over GF(11) for k ≤ 7 and QC codes over GF(13) for k ≤ 6. Venkaiah and Gulliver [31] constructed quasi-cyclic [pk,k,d]13 codes of dimensions k ≤ 6 and n ≤ 150. In [12] and [13] E. Chen and N. Aydin constructed 45 new OC and QT codes over GF(11), 38 such codes over GF(13) and presented databases for small dimensions and n ≤ 150. New QT codes over GF(11) and GF(13) are also presented in [7]. Three-dimensional projective codes are closely related to (n,r)−arcs in projective finite planes, and a lot of research has been done over finite fields of size up to 19 [8]. Recently an optimal (78, 8)-arc in PG(2,11) was constructed in [25] as a [78, 3, 70]11 QC code. In this paper we present 34 new QC codes over GF(11) and 36 new QC codes over GF(13). 2. QC codes A code C is said to be p-QC if a cyclic shift of any codeword by p positions results in another codeword. Suppose that C is a p-QC [pm,k] code (m ≥ k). It is convenient to take the co-ordinate places of C in the following order 1,p + 1, 2p + 1, . . . , (m− 1)p + 1, 2,p + 2, . . . , (m− 1)p + 2, p, 2p,. . . ,mp. Then C will be generated by a matrix of the form [B1,B2, . . . ,Bp] where each Bi is a circulant matrix, i.e. a matrix of the form B =   b0 b1 b2 · · · bm−1 bm−1 b0 b1 · · · bm−2 bm−2 bm−1 b0 · · · bm−3 ... ... ... ... b1 b2 b3 · · · b0   If the row vector (b0b1 · · ·bm−1) is identified with the polynomial d(x) = b0 + b1x + ... + bm−1xm−1, then we may write 238 R. Daskalov, E. Metodieva / J. Algebra Comb. Discrete Appl. 7(3) (2020) 237–245 B =   d(x) xd(x) x2d(x) ... xm−1d(x)   where each polynomial is reduced modulo xm − 1. Denote the polynomials associated with the matrices B1, B2, B3, . . . , Bp by d1(x), d2(x), d3(x), . . . , dp(x). These polynomials are called defining polynomials of C. Taking the polynomials axldi(x) instead of di(x) we make a cyclic shift of the columns of Bi and multiply them by a nonzero element of the field. This leads to a generator matrix of an equivalent code. So, the defining polynomials of a QC-code can be chosen from a fixed set of representatives of the equivalence classes of polynomials of degree less than m under the following relation: ci(x) ≈ cj(x) ⇐⇒ ci(x) ≡ axlcj(x) mod (xm − 1) (1) It stands to reason that we need an efficient algorithm to produce such a set of polynomials. 3. Necklaces We identify the polynomials with the strings of their coefficients. In terms of strings the relation (1) is a composition of two actions on strings, namely, rotating the string and multiplying its entries by nonzero elements of the field (scaling the string). Efficient algorithms are known for generating the equivalence classes of the first of the actions. By efficient we mean that the amount of computation used in generating the objects is proportional to the number of objects generated. Let Σq be the alphabet {0, 1, 2, . . . ,q − 1} and Σmq be the set of q-ary strings of length m. Denote by αi, 1 ≤ i ≤ m, the entries of the string α ∈ Σmq , and at is the string of length t whose entries are all equal to a. The symbol � is used for lexicographic order in Σmq . Call two strings equivalent if one is a cyclic shift of the other. An equivalence class of strings under this relation is called a necklace of m beads in q colors. We identify each necklace with the lexicographically smallest representative in the equivalence class. Thus we call a string α = α1α2 . . .αm a necklace if, for 1 ≤ i ≤ m, α1 . . .αm � αi . . .αmα1 . . .αi−1. For given m and q the number of necklaces is well known to be Nq(m) = 1 m ∑ d|m φ(d)q m d where φ is the Euler totient function. A simple and elegant algorithm was proposed by Fredricksen, Kessler and Maiorana [22], [23] to generate all necklaces in Σmq . We will refer to this as the FKM algorithm. For given m and q, the FKM algorithm creates a list, FKM(q,m), consisting of a certain subset of Σmq in lexicographic order. The list begins with the string 0 m and ends with (q − 1)m. For a given α 239 R. Daskalov, E. Metodieva / J. Algebra Comb. Discrete Appl. 7(3) (2020) 237–245 on FKM(q,m), the successor of α, succ(α), is obtained as follows: For α = α1α2 . . .αm ≺ (q − 1)m, succ(α) = (α1 . . .αi−1(αi + 1))tα1 . . .αj, where i is the largest integer 1 ≤ i ≤ m such that αi < q − 1 and t,j satisfy ti + j = m, j < i. It is shown in [22] that there is no necklace between two elements of FKM(q,m), so that the list contains all necklaces. Thus, discarding non necklaces of FKM(q,m) would result in a list of all necklaces in increasing order. In [29] Ruskey, Savage and Wang proved that succ(α) is a necklace if and only if the ”i” from the definition of the successor of α is a divisor of m. Including this test, the entire algorithm can be represented by the following PASCAL code: for i:=0 to m do a[i]:=0; br:=0; i:=m; repeat a[i]:=a[i]+1; for j:=1 to m-i do a[j+i]:=a[j]; if m mod i = 0 then begin br:=br+1; for j:=1 to m do write(a[j]); writeln; end; i:=m; while a[i]=q-1 do i:=i-1; until i=0; 4. Generalized necklaces In its turn, scaling the necklaces partitions the set of all necklaces into new equivalence classes. We call them generalized necklaces. To generate their representatives we know no more efficient algorithm than rejecting those necklaces that are not the smallest representatives. A naive approach to testing whether a length m necklace is the smallest representative of an equivalence class according to (1) is to compare the necklace with all of its scaled rotations. However, by taking into consideration some facts we can decrease the number of comparisons that have to be made. First of all we generate only necklaces with first non-zero element 1. It follows from the FKM algorithm that any necklace has the form α = (α1α2 . . .αi) t, t ≥ 1 Any rotation of α has the form α′ = αs+1 . . .αi(α1α2 . . .αi) t−1α1 . . .αs = = (αs+1 . . .αiα1 . . .αs) t for s = 1, 2, . . . , i− 1. Thus comparing the strings α and bα′, b ∈ GF(q) \ {0, 1} it suffices to compare the substring β = α1α2 . . .αi with its rotations, multiplied by an element of GF(q) \{0, 1}. If p is the index of the first non-zero element of β, than scaled cyclic shifts with starting positions 2, 3, . . . ,p will obviously follow β in lexicographic order. The cyclic shifts with starting positions i−p + 2, i−p + 3, . . . , i 240 R. Daskalov, E. Metodieva / J. Algebra Comb. Discrete Appl. 7(3) (2020) 237–245 will have αi 6= 0 in position with number less than p and therefore they will also follow β = α1 . . .αi. So, the only comparisons that have to be made are with cyclic shifts starting from positions l = p + 1,p + 2, . . . , i−p + 1 if i−p + 1 ≥ p + 1, i.e. i ≥ 2p, having αl+p−1 > 1, and multiplied only by the inverse of αl+p−1. The implementation of these considerations yields the following PASCAL code, which produces the desired set of strings (or polynomials): for i:=0 to n-1 do a[i]:=0; a[n]:=1; br:=1; i:=n; p:=n; repeat if i=p then begin i:=i-1; p:=p-1; end; a[i]:=a[i]+1; for j:=1 to n-i do a[j+i]:=a[j]; if n mod i = 0 then begin for l:=p+1 to i-p+1 do if a[l+p-1]>1 then begin m:=inv[a[l+p-1]]; k1:=1; k2:=l; while (k1<=i) and (a[k1]=mul[a[k2],m]) do begin k1:=k1+1; k2:=(k2 mod i)+1; end; if k1<= i then if a[k1] > mul[a[k2],m] then goto 10; end; begin br:=br+1; for j:=1 to n do write(a[j]); end; end; 10: i:=n; while a[i]=q-1 do i:=i-1; until (i=i1) and (i=1); In the table below the number K11(m) and K13(m) of generated objects is given for m = 6, 7, 8, 9. The respective number N11(m) and N13(m) of necklaces is also given for comparison. From some seconds up to some minutes are needed for the generation of the objects (CPU, Intel i3, 2.2GHz). Table 1: Nq(m) and Kq(m) m N11(m) K11(m) m N13(m) K13(m) 7 2783891 278389 6 804895 67116 8 26796726 2679859 7 8964085 747007 9 261994491 26199449 8 101969959 8497806 241 R. Daskalov, E. Metodieva / J. Algebra Comb. Discrete Appl. 7(3) (2020) 237–245 The next explicit formula for the number of generalized necklaces (defining polynomials for QC codes) was given recently in [31] (for QT codes see [32]) Kq(m) = 1 (q − 1)m ∑ d|m φ(d) ( q m d − 1 ) gcd(d,q − 1) 5. New QC codes over GF(11) and GF(13) In this section we present 34 new QC codes over GF(11) in dimensions k = 8, 9 and 36 new QC codes over GF(13) in dimensions k = 7, 8. The new codes are obtained by non-exhaustive local computer search. By reason of space the defining polynomials and weight distributions only of some of the codes are given. For the rest of the codes, the respective information is available on request from the authors. All constructed codes will be send to E. Chen to be included in the respective table[9]. The elements of the fields are denoted by 0, 1, 2, ... , 9, 10 = a, 11 = b, 12 = c. Codes over GF(11) There exist quasi-cyclic codes with parameters: [16, 8, 8]11, [24, 8, 14]11, [32, 8, 20]11, [40, 8, 26]11, [48, 8, 33]11, [56, 8, 39]11, [64, 8, 46]11, [72, 8, 53]11, [80, 8, 59]11, [88, 8, 66]11, [96, 8, 73]11, [104, 8, 80]11, [112, 8, 87]11, [120, 8, 94]11, [128, 8, 100]11, [136, 8, 107]11, [144, 8, 114]11, [152, 8, 121]11. A [16, 8, 8]11 optimal code: aaaaaaa9, aaa9a362; 01 87400 955200 10367360 112031680 128526000 1326073600 1456016800 1574628160 1646652680 A [24, 8, 14]11 code: aaa986a2, aaaa9a68, aaaa7022; 01 144880 1529840 16159390 17753120 182924120 199254640 2023128500 2144045200 2260063480 2352233280 2421762430 A [32, 8, 20]11 code: aaa98558, aaaa6960, aaaa9784, a9073007; 01 202540 2113360 2266000 23280960 241077200 253415120 269184520 2720465280 2836499940 2950328800 3050421000 3132438240 3210165920 A [40, 8, 26]11 code: aaa98503, aaaa7006, aaaa9a28, a9074154, aaaa9232; 01 261360 274880 2824300 29113760 30406400 311287440 323650050 338810720 3418211080 3531126480 3643291000 3746825520 3836935880 3918923760 404746250 A [48, 8, 33]11 code: a9074154, aaa986a2, aaaa9232, aaaa7022, aaaa9a68, a92a7a64; 01 332400 3410200 3544800 36152880 37501200 381448040 393704240 408343150 4116214080 4227158960 4337882960 4442926260 4538227600 4624916320 4710623840 482201950 A [56, 8, 39]11 code: a9074154, aaa98449, aaaa7011, aaaa9a28, aaaa9162, aa461627, aaa80096; 01 391120 403720 4116160 4263720 43192000 44575440 451534240 463666320 477828560 4814643050 4923845680 5033507360 5139412640 5237813820 5328560000 5415893160 555777200 561024690 A [64, 8, 46]11 code: aaa98449, aa606690, aaaa97a4, a9072196, aaaa9149, aa461627, aaa80091, aa2a9a89; 01 462000 476640 4823420 4974960 50234280 51628720 521577940 533584880 547284360 5513222240 5621322560 5729816640 5836078240 5936648480 6030601420 6119981760 629712480 633080160 64477700 A [72, 8, 53]11 code: aaa986a2, aaaa7022, aaaa9a68, a9074154, aaaa9232, aa461651, aaa800a5, aa301291, aa6a5912; 01 532480 549760 5529760 5693600 57258960 58665280 591602880 603442340 616787200 6212060720 6319044240 6426783290 6533069280 6635145400 6731353600 6823078000 6913339120 705749880 711620720 72222370 A [80, 8, 59]11 code: a8606780, aaaa6a63, aa8896a9, a9074154, aaa25520, aa461651, aaa7a862, aa2a9a89, aa6a5384, aa3a2803; 01 591200 603520 6112080 6240120 63102640 64283920 65691600 661585440 673298320 686291980 6910963040 7017246760 7124241840 242 R. Daskalov, E. Metodieva / J. Algebra Comb. Discrete Appl. 7(3) (2020) 237–245 7230352890 7333273280 7431397080 7525181760 7616544120 778602400 783299480 79839040 80106370 A [88, 8, 66]11 code: a8606759, aaaa6a60, aa8896a6, a9074150, aaa255a5, aa461649, aaa7a857, aa2a9a86, aa6a5379, aa3a2803, aaa97887; 01 661440 674960 6816000 6942400 70118120 71296160 72698300 731567200 743124800 755893520 7610031240 7715640320 7821953920 7927887520 8031397440 8131029840 8226457160 8319099600 8411413680 855327520 861884000 87424960 8848780 There exist quasi-cyclic codes with parameters: [9p, 9, 7p− 6]11 for p = 2, 3, ..., 7, [9p, 9, 15p/2 − 9]11 for p = 8, 10, 12, 14, 16 and [9p, 9, 15(p− 1)/2 − 2]11 for p = 9, 11, 13, 15, 17. A [18, 9, 8]11 code: aaaa98708, aaaaa7023; 01 81440 922320 10187740 111340640 127871640 1336363600 14129741480 15346074720 16648874890 17763370460 18424098760 A [45, 9, 29]11 code: aaaaa7023, aaaaa9949, aaaa98708, aa98113a9, aaa9a9a68; 01 292880 3011160 3153550 32236160 33930270 343294720 3510311930 3628639260 3769744510 38146759040 39263601600 40395301510 41481704840 42459168780 43320269140 44145588050 4532330290 A [108, 9, 81]11 code: aaaa98708, aaaaa7023, aaaaa9949, aa98113a9, aaa9a9a59, aaa520467, aaa249519, aa9a9a937, aaa721a86, aa9a9a823, aaa9a8973, aa9a89a82 01 811800 825400 8317280 8450580 85142650 86394740 87976260 882353500 895258880 9011138730 9121897720 9240764510 9369821070 94111537090 95164458260 96222387870 97275444100 98309390300 99312102790 100280789380 101222478830 102152968320 10388963020 10442785460 10516267290 1064608720 107864810 10878330 Codes over GF(13) There exist quasi-cyclic codes with parameters: [14, 7, 7]13, [21, 7, 13]13, [28, 7, 18]13, [7p, 7, 6p − 6]13 for p = 5, 6, ..., 15 and [7p, 7, 6p− 5]13 for p = 16, 17, 18, 19. A [14, 7, 7]13 code: ccccccb, cccbca7; 01 72100 821336 9164220 10983556 114319196 1212922308 1323875068 1420460732 A [21, 7, 13]13 code: cc484c1, ca650c9, ccb0b70; 01 136636 1439396 15210336 16957684 173356220 189025884 1917011344 2020458452 2111682564 A [28, 7, 18]13 code: cc48499, ca650c7, cc78aca, cccca17; 01 181848 197896 2049560 21219168 22847980 232627268 246584004 2512663000 2617505348 2715570492 286671952 A [35, 7, 24]13 code: cc4849a, ca650c9, cc78acb, cccca15, cb04650; 01 242772 2511172 2652500 27209160 28713412 292061696 304989516 319623040 3214413560 3315735888 3411134032 353801768 A [42, 7, 30]13 code: cc48499, cc78b00, cccca18, cb04647, ccbb381, ca650c9; 01 302352 3112768 3251912 33194376 34599592 351619184 363827880 377441812 3811757564 3914433048 4013019076 417613844 422175108 A [112, 7, 91]13 code: cccca24, ca650c9, cc484c1, ccc76a8, ccbc4b2, ccbb385, cccbcb4, cc78b01, ccb9656, ccbc163, ccbc962, ccbac74, ccbab1a, ccbab23, ccccbc1, cb38697; 01 914200 9214448 9335784 9491056 95196224 96407736 97825636 981507632 992539824 1003945984 1015683272 1027303128 1038580348 1048870568 1058095248 1066441624 1074324320 1082388792 1091068228 110336672 11179716 1128076 There exist quasi-cyclic codes with parameters: [16, 8, 8]13, [24, 8, 14]13, [32, 8, 20]13, [40, 8, 27]13, [48, 8, 33]13, [56, 8, 40]13, [64, 8, 47]13, [72, 8, 53]13 and [8p, 8, 7p− 10]13 for p = 10, 11, ..., 19. A [16, 8, 8]13 code: cccccc77, cc82b0ca; 01 87116 980352 10583632 114004448 1219892040 1373427424 14188896848 15302188704 16226650156 243 R. Daskalov, E. Metodieva / J. Algebra Comb. Discrete Appl. 7(3) (2020) 237–245 A [24, 8, 14]13 code: ccccccc6, ccb50a09, ccc538ca; 01 143312 1532544 16205248 171154400 185379504 1920390304 2061267056 21139904640 22228977184 23238950528 24119466000 A [64, 8, 47]13 code: ccccccca, ccb50a15, ccc7b8b6, ccb75441, cc629b0b, cc847138, ccc538ca, cc6c9aa0; 01 473168 4812156 4950016 50189360 51593184 521788792 534855392 5411944992 5525823712 5650130300 5784293952 58122167440 59148750176 60149087808 61117161664 6268071872 6325970304 644836432 A [80, 8, 60]13 code: cccccc56, ccb50968, ccc538c8, ccc7c166, ccb75438, cc629a31, cc8471a8, cc6c9a95, cccb9586, ccb57b73; 01 601584 613840 6219200 6362112 64199032 65583776 661584240 673965664 689137136 6919061568 7035892528 7160720384 7291043424 73119819040 74135862176 75130555584 76102976704 7764252800 7829645280 798981952 801362696 References [1] N. Aydin, I. Siap, D. K. Ray–Chaudhuri, The structure of 1–generator quasi–twisted codes and new linear codes, Des. Codes Cryptogr. 24 (2001) 313–326. 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