ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.645015 J. Algebra Comb. Discrete Appl. 7(1) • 3–20 Received: 27 June 2019 Accepted: 20 September 2019 Journal of Algebra Combinatorics Discrete Structures and Applications Construction of quasi-twisted codes and enumeration of defining polynomials Research Article T. Aaron Gulliver, Vadlamudi Ch. Venkaiah Abstract: Let dq(n,k) be the maximum possible minimum Hamming distance of a linear [n,k] code over Fq. Tables of best known linear codes exist for small fields and some results are known for larger fields. Quasi-twisted codes are constructed using m×m twistulant matrices and many of these are the best known codes. In this paper, the number of m × m twistulant matrices over Fq is enumerated and linear codes over F17 and F19 are constructed for k up to 5. 2010 MSC: 94B05, 94B65, 94A55 Keywords: Finite fields, Twistulant matrices, Quasi-twisted codes, Optimal codes, Griesmer bound 1. Introduction Let Fq denote the finite field of q elements, and V (n,q) the vector space of n-tuples over Fq. A linear [n,k] code C of length n and dimension k over Fq is a k-dimensional subspace of V (n,q). The elements of C are called codewords. The (Hamming) weight of a codeword is the number of non-zero coordinates, and the minimum distance of C is the smallest weight among all non-zero codewords of C. An [n,k,d] code is an [n,k] code with minimum distance d. Let Ai be the number of codewords of weight i in C. Then the numbers A0,A1, . . . ,An are called the weight distribution of C. A central problem in coding theory is that of optimizing one of the parameters n,k and d for given values of the other two. One can find dq(n,k), the largest value of d for which there exists an [n,k,d] code over Fq, or nq(k,d), the smallest value of n for which there exists an [n,k,d] code over Fq. A code which achieves either of these values is called optimal. Tables of best known linear codes exist for q = 2 to 9 [6], 11 [3] and 13 [4]. T. Aaron Gulliver (Corresponding Author); Department of Electrical and Computer Engineering, University of Victoria, PO Box 1700, STN CSC, Victoria, BC V8W 2Y2, Canada (email: agullive@ece.uvic.ca). Vadlamudi Ch. Venkaiah; School of Computer and Information Sciences, University of Hyderabad, Gachibowli, Hyderabad 500 046, India (email: venkaiah@hotmail.com, vvcs@uohyd.ernet.in) 3 https://orcid.org/0000-0001-9919-0323 https://orcid.org/0000-0001-5440-0200 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 The Griesmer bound is a well-known lower bound on nq(k,d) nq(k,d) ≥ gq(k,d) = k−1∑ j=0 ⌈ d qj ⌉ , (1) where dxe denotes the smallest integer ≥ x. For k ≤ 2, there exist codes that attain equality in the Griesmer bound for all q and d. The Singleton bound [12] is a lower bound on nq(k,d) and is given by nq(k,d) ≥ d + k −1. (2) Codes that meet this bound are called maximum distance separable (MDS). MDS codes exist for all values of n ≤ q + 1. Thus, for q = 17 MDS codes exist for all lengths 18 or less, and for q = 19 all lengths 20 or less. Note that all MDS codes are optimal. For larger lengths and dimensions, far less is known about codes over F17 and F19. MDS self-dual codes (k = n/2), of lengths 2, 4, 6, 8, 10 and 18 over F17 are known [1], as well as self-dual [12,6,6], [14,7,7], [16,8,8], [20,10,10], [22,11,10] and [24,12,10] codes. The [14,7,8] and [20,10,10] extended quadratic residue (QR) codes are given in [13]. Using Magma [2], it was determined that the next extended QR code has parameters [44,22,18]. MDS self-dual codes of lengths 4, 8, 12 and 20 over F19 are known [1], as well as self-dual [16,8,8] and [24,12,9] codes. The [18,9,9] and [32,16,14] extended QR codes are given in [13]. In this paper, codes over F17 and F19 for k up to 5 are presented. These codes establish lower bounds on the minimum distance. Many of these meet the Singleton and/or Griesmer bounds, and so are optimal. 2. Quasi-twisted codes A constacyclic shift of an m-tuple (x0,x1, . . . ,xm−1), is the m-tuple (λxm−1,x0, . . . ,xm−2), where λ ∈ Fq\{0}, and a constacyclic shift by p positions is the m-tuple (λxm−p, . . . ,λxm−1,x0, . . . ,xm−p−1). A linear code C is said to be quasi-twisted (QT) if a constacyclic shift of any codeword by p positions is also a codeword in C [8]. Note that quasi-twisted codes generalize the classes of constacyclic codes (p = 1), quasi-cyclic codes (λ = 1), cyclic codes (λ = 1, p = 1), and negacyclic codes (λ = −1, p = 1). The length of a QT code considered here is n = mp. With a suitable permutation of coordinates, many QT codes can be characterized in terms of m×m twistulant matrices. In this case, a QT code can be transformed into an equivalent code with generator matrix G = [B0 B1 B2 . . . Bp−1] , (3) where Bi, i = 0,1, . . . ,p− 1, is an m×m twistulant matrix (also known as a constacyclic matrix), over Fq of the form [9] B =   b0 b1 b2 · · · bm−1 λbm−1 b0 b1 · · · bm−2 λbm−2 λbm−1 b0 · · · bm−3 ... ... ... ... λb1 λb2 λb3 · · · b0   , 4 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 where λ ∈ Fq\{0} and bi,0 ≤ i ≤ m − 1, are elements of Fq. When λ = 1, a twistulant matrix is a circulant matrix, and when λ = −1, a twistulant matrix is known as a negacirculant matrix [8]. The algebra of m×m twistulant matrices over Fq is isomorphic to the algebra of polynomials in the ring Fq[x]/(xm −λ) if Bi is mapped onto the polynomial bi(x) = b0,i + b1,ix + b2,ix2 + . . . + bm−1,ixm−1 formed from the entries in the first row of Bi. The bi(x) associated with a QT code are called the defining polynomials [7]. The set {b0(x),b1(x), . . . ,bp−1(x)} defines an [mp,m] QT code with k ≤ m. 3. Defining polynomials The construction of QT codes requires a representative set of defining polynomials. These are the equivalence class representatives of a partition of the set of polynomials of degree less than m. For defining polynomials, multiplication by a non-zero element of Fq does not change the weight and hence does not change the equivalence class. Thus, two polynomials rj(x) and ri(x) are said to be equivalent if rj(x) = γx lri(x) mod (x m −λ), for some integer l ≥ 0 and scalar γ ∈ Fq\{0}. A closed-form expression for the number of defining polynomials is now given. Let g be the permu- tation (1,2, . . . ,m) so that g maps i to i + 1, for 1 ≤ i ≤ m− 1, and m to 1. Therefore, gi, 1 ≤ i ≤ m, is also a permutation and has order m gcd(m,i) in the symmetric group of degree m. Thus, the action of g on the m-tuple x = (x1,x2, . . . ,xm) changes x to (xm,x1,x2, . . . ,xm−1) where xi ∈ Fq. Now let λg be such that the action of λg on the m-tuple changes x to (λxm,x1,x2, . . . ,xm−1), the action of (λg)2 on x results in (λxm−1,λxn,x1,x2, . . . ,xm−2), and similarly for other powers. Then the order of λg is Ord(λ)m, where Ord(λ) is the order of λ in Fq. Further, let t(λg), t ∈ Fq \{0}, be such that it changes x to (tλxm, tx1, tx2, . . . , txm−1). The action of t(λg)2 changes x to (tλxm−1, tλxm, tx1, tx2, . . . , txm−2), and similarly for other powers. The equivalence relation is induced by the action of the group consisting of the elements t(λg)i, 1 ≤ i ≤ Ord(λ)m, t ∈ Fq \{0}. Distinct equivalence classes correspond to distinct orbits under the action of this group and so can be enumerated using Burnside’s Lemma [5, 9]. Definition 3.1. An ordered m-tuple (or word of length m), x = (x1,x2, . . . ,xm), is said to be fixed by t(λg)i, t,λ ∈ Fq \{0}, 1 ≤ i ≤ Ord(λ)m, if the m-tuple x remains unchanged by the action of t(λg)i. Theorem 3.2. The number of words of length m over the alphabet Fq fixed by t(λg)i for some fixed λ ∈ Fq \{0}, t ∈ Fq \{0}, 1 ≤ i ≤ Ord(λ)m, is qgcd(m,i) if t ( m gcd(m,i) ) λ ( i gcd(m,i) ) = 1. Otherwise, it is 1. Proof. Let x = (x1,x2, . . . ,xm) be a word of length m over Fq. Then the relation between the components of x before and after the action of t(λg)i is x1 = tλxm−i+1 x2 = tλxm−i+2 x3 = tλxm−i+3 ... xi−1 = tλxm−1 xi = tλxm xi+1 = tx1 xi+2 = tx2 ... xm−1 = txm−i−1 xm = txm−i 5 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 if 1 ≤ i ≤ m. If m + 1 ≤ i ≤ 2m, the relation between the components of x is x1 = tλ 2xm−i+1 x2 = tλ 2xm−i+2 x3 = tλ 2xm−i+3 ... xi−1 = tλ 2xm−1 xi = tλ 2xm xi+1 = tλx1 xi+2 = tλx2 ... xm−1 = tλxm−i−1 xm = tλxm−i. Thus in general, the relation between the components of x is x1 = tλ jxm−i+1 x2 = tλ jxm−i+2 x3 = tλ jxm−i+3 ... xi−1 = tλ jxm−1 xi = tλ jxm xi+1 = tλ j−1x1 xi+2 = tλ j−1x2 ... xm−1 = tλ j−1xm−i−1 x = tλj−1xm−i if (j −1)m + 1 ≤ i ≤ jm, 1 ≤ j ≤ Ord(λ). Let gcd(m,i) = h. Then, from the expressions above, the orbit of xm is xm = txm−i = t 2xm−2i = · · · = tb m i cxm−bm i ci = t(b m i c+1)λx2m−(bm i c+1)i = t (bm i c+2)λx2m−(bm i c+2)i = · · · = t(b 2m i c−1)λx2m−(b2m i c−1)i = t(b 2m i c)λx2m−(b2m i c)i = t (b2m i c+1)λ2x3m−(b2m i c+1)i = t(b 2m i c+2)λ2x3m−(b2m i c+2)i = · · · = t (b3m i c−1)λ2x3m−(b3m i c−1)i = t(b 3m i c)λ2x3m−(b3m i c)i = t (b3m i c+1)λ3x4m−(b3m i c+1)i = t(b 3m i c+2)λ3x4m−(b3m i c+2)i = · · · = t(b 4m i c−1)λ3x4m−(b4m i c−1)i = t(b 4m i c)λ3x4m−(b4m i c)i = · · · = t (b( i h −1) m i c+1)λ( i h −1)x im h −(b ( i h −1)m i c+1)i = t(b( i h −1) m i c+2)λ( i h −1)x im h −(b ( i h −1)m i c+2)i = · · · = t(b( i h m i )c−1)λ( i h −1)x im h −(b im h i c−1)i = t(b( i h m i )c)λ( i h )xm, 6 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 and so we also have xm−1 = txm−1−i = t 2xm−1−2i = · · · = tb m i cxm−1−bm i ci = t(b m i c+1)λx2m−1−(bm i c+1)i = t (bm i c+2)λx2m−1−(bm i c+2)i = · · · = t(b 2m i c−1)λx2m−1−(b2m i c−1)i = t (b2m i c)λx2m−1−(b2m i c)i = t(b 2m i c+1)λ2x3m−1−(b2m i c+1)i = t (b2m i c+2)λ2x3m−1−(b2m i c+2)i = · · · = t(b 3m i c−1)λ2x3m−1−(b3m i c−1)i = t (b3m i c)λ2x3m−1−(b3m i c)i = t(b 3m i c+1)λ3x4m−1−(b3m i c+1)i = t (b3m i c+2)λ3x4m−1−(b3m i c+2)i = · · · = t(b 4m i c−1)λ3x4m−1−(b4m i c−1)i = t (b4m i c)λ3x4m−1−(b4m i c)i = · · · = t(b( i h −1) m i c+1)λ( i h −1)x im h −1−(b ( i h −1)m i c+1)i = t(b( i h −1) m i c+2)λ( i h −1)x im h −1−(b ( i h −1)m i c+2)i = · · · = t(b( i h m i )c−1)λ( i h −1)x im h −1−(b im h i c−1)i = t(b( i h m i )c)λ( i h )xm−1. Similar expressions exist for xm−2,xm−3, . . . ,xm−h+2, and in general xm−h+1 = txm−h+1−i = t 2xm−h+1−2i = · · · = tb m i cxm−h+1−bm i ci = t(b m i c+1)λx2m−+1−(bm i c+1)i = t (bm i c+2)λx2m−h+1−(bm i c+2)i = · · · = t(b 2m i c−1)λx2m−h+1−(b2m i c−1)i = t (b2m i c)λx2m−h+1−(b2m i c)i = t(b 2m i c+1)λ2x3m−h+1−(b2m i c+1)i = t (b2m i c+2)λ2x3m−h+1−(b2m i c+2)i = · · · = t(b 3m i c−1)λ2x3m−h+1−(b3m i c−1)i = t (b3m i c)λ2x3m−h+1−(b3m i c)i = t(b 3m i c+1)λ3x4m−h+1−(b3m i c+1)i = t (b3m i c+2)λ3x4m−h+1−(b3m i c+2)i = · · · = t(b 4m i c−1)λ3x4m−h+1−(b4m i c−1)i = t (b4m i c)λ3x4m−h+1−(b4m i c)i = · · · = t(b( i h −1) m i c+1)λ( i h −1)x im h −h+1−(b ( i h −1)m i c+1)i = t(b( i h −1) m i c+2)λ( i h −1)x im h −h+1−(b ( i h −1)m i c+2)i = · · · = t(b( i h m i )c−1)λ( i h −1)xim h −h+1−( im ih −1)i = t( i h m i )λ( i h )xm−h+1. Thus, the orbits of xm, xm−1, . . ., xm−h+1 are fixed by the action of t(λg)i if and only if t m h λ i h = t m gcd(m,i) λ i gcd(m,i) = 1. Since there are h = gcd(m,i) independent orbits and each orbit can take on q values, qh = qgcd(m,i) words are fixed by t(λg)i. If t m gcd(m,i) λ i gcd(m,i) 6= 1, then there is only one orbit consisting of all m elements of the word. In this case, since λ 6= 1, the only word that is fixed is the zero word. Theorem 3.3. The number of defining polynomials of length m over Fq is Mq,λ(m) = 1 (q −1)Ord(λ)m Ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 (qgcd(m,i) −1) + 1 (4) 7 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Proof. There are (q − 1)Ord(λ)m permutations given by t(λg)i. Thus, by Burnside’s lemma [5], the number of orbits of words of length m over an alphabet of size q is equal to the average number of words fixed by each t(λg)i, 1 ≤ i ≤ Ord(λ)m, t ∈ Fq \{0}. Therefore, we have Mq,λ(m) = 1 (q −1)Ord(λ)m Ord(λ)m∑ i=1 t∈Fq\{0} |Fix t(λg)i|, where |Fix t(λg)i| denotes the number of words fixed by t(λg)i. From Theorem 1, the number of words fixed by t(λg)i is either qgcd(m,i) or 1 depending on whether t m gcd(m,i) λ i gcd(m,i) = 1 or not. Therefore Mq,λ(m) = 1 (q −1)Ord(λ)m   Ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 qgcd(m,i) +   Ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) 6=1 1     = 1 (q −1)Ord(λ)m   Ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 qgcd(m,i) +  (q −1)Ord(λ)m− Ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 1     = 1 (q −1)Ord(λ)m   Ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 (qgcd(m,i) −1) + (q −1)Ord(λ)m   = 1 (q −1)Ord(λ)m Ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 (qgcd(m,i) −1) + 1 Example 3.4. Let q = 13, m = 2, and λ = 4. Since 6 is the least integer such that λ6 = 46 = 1, we 8 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 have Ord(λ) = Ord(4) = 6. Then M13,4(2) = 1 144 12∑ i=1 t∈F13\{0},t 2 gcd(2,i) 4 i gcd(2,i) =1 (13gcd(2,i) −1) + 1 = 1 144 {[ (13gcd(2,12) −1) ] + [ (13gcd(2,5) −1) + (13gcd(2,11) −1) ]} + 1 144 {[ (13gcd(2,8) −1) ] + [ (13gcd(2,10) −1) ]} + 1 144 {[ (13gcd(2,3) −1) + (13gcd(2,9) −1) ] + [ (13gcd(2,1) −1) + (13gcd(2,7) −1) ]} + 1 144 {[ (13gcd(2,1) −1) + (13gcd(2,7) −1) ]} + 1 144 {[ (13gcd(2,3) −1) + (13gcd(2,9) −1) ]} + 1 144 {[ (13gcd(2,4) −1) ] + [ (13gcd(2,2) −1) ]} + 1 144 {[ (13gcd(2,5) −1) + (13gcd(2,11) −1) ] + [ (13gcd(2,6) −1), ]} + 1. because 1 2 gcd(2,12) 4 12 gcd(2,12) = 2 2 gcd(2,5) 4 5 gcd(2,5) = 2 2 gcd(2,11) 4 11 gcd(2,11) = 3 2 gcd(2,8) 4 8 gcd(2,8) = 4 2 gcd(2,10) 4 10 gcd(2,10) = 5 2 gcd(2,3) 4 3 gcd(2,3) = 5 2 gcd(2,9) 4 9 gcd(2,9) = 6 2 gcd(2,1) 4 1 gcd(2,1) = 6 2 gcd(2,7) 4 7 gcd(2,7) = 7 2 gcd(2,1) 4 1 gcd(2,1) = 7 2 gcd(2,7) 4 7 gcd(2,7) = 8 2 gcd(2,3) 4 3 gcd(2,3) = 8 2 gcd(2,9) 4 9 gcd(2,9) = 9 2 gcd(2,4) 4 4 gcd(2,4) = 10 2 gcd(2,2) 4 2 gcd(2,2) = 11 2 gcd(2,5) 4 5 gcd(2,5) = 11 2 gcd(2,11) 4 11 gcd(2,11) = 12 2 gcd(2,6) 4 6 gcd(2,6) = 1, and therefore M13,4(2) = 1 144 [(132 −1) + [(13−1) + (13−1)] + (132 −1) + (132 −1) + [(13−1) + (13−1)]] + 1 144 [[(13−1) + (13−1)] + [(13−1) + (13−1)] + [(13−1) + (13−1)]] + 1 144 [(132 −1) + (132 −1) + [(13−1) + (13−1)] + (132 −1)] + 1 = 9. Note that setting λ = 1 in (4) gives the number of defining polynomials for quasi-cyclic codes [14] Mq,1(m) = 1 (q −1)m ∑ i|m φ(i) gcd(i,q −1)(qm/i −1) + 1. (5) Table 1 gives the number of defining polynomials over F2, F3, F5, F7, and F11 with λ = 1, Table 2 gives the number of defining polynomials over F4, F8, F9, and F16 with λ = 1, and Table 3 gives the number of defining over F13, F17, and F19 with λ = 1. To illustrate the effect of λ, Tables 4 and 5 give the number of defining polynomials over F3 and F4. 9 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Table 1. The number of defining polynomials over F2, F3, F5, F7, and F11 with λ = 1 q m 2 3 5 7 11 1 2 2 2 2 2 2 3 4 5 6 8 3 4 6 12 22 46 4 6 14 45 106 374 5 8 26 158 562 3226 6 14 68 665 3298 29576 7 20 158 2792 19610 278390 8 36 424 12255 120206 2679860 9 60 1098 54262 747330 26199450 10 108 2980 244301 4708486 259377496 11 188 8054 1109732 29959498 2593742462 12 352 22218 5086965 192243598 26153599626 Table 2. The number of defining polynomials over F4, F8, F9, and F16 with λ = 1 q m 4 8 9 16 1 2 2 2 2 2 4 6 7 10 3 10 26 32 94 4 24 150 213 1098 5 70 938 1478 13986 6 238 6258 11107 186478 7 782 42806 85412 2556530 8 2744 299670 672825 35791946 9 9726 2130458 5380862 509033346 10 34990 15339642 43586287 7330084546 11 127102 111557594 356602952 106619309362 12 466198 818092242 2941985613 1563749966062 4. Quasi-twisted codes over F17 and F19 In this section, the defining polynomials given above are used to construct quasi-twisted codes over F17 and F19. The number of defining polynomials over F17 for m = 1 to 5 is given in Table 6 and over F19 for m = 1 to 5 in Table 7. Note that the zero polynomial is not considered in constructing codes. Considering a code structure (i.e. QT), results in a search space that is smaller than for the general code design problem. The more restrictions on the structure, the smaller the search, but this creates a tradeoff since good codes may be missed if too much structure is imposed on the code. The QT codes presented here were constructed using a stochastic optimization algorithm, namely tabu search, which is similar to that in [10, 11, 14]. By restricting the search for good codes to the class of QT codes, and using a stochastic heuristic, codes with high minimum distance can be found with a reasonable amount of computational effort. Based on the results obtained here, this approach provides a good tradeoff. The search for a (pm,m) QT code begins with a random set of p defining polynomials. A polynomial 10 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Table 3. The number of defining polynomials over F13, F17, and F19 with λ = 1 q m 13 17 19 1 2 2 2 2 9 11 12 3 64 104 130 4 605 1317 1822 5 6190 17750 27514 6 67117 251543 435760 7 747008 3663740 7094222 8 8497807 54499433 117943232 9 98189934 823526990 1991899630 10 1148826961 12599979635 34061506732 11 13576972684 194726683568 588334640902 12 161792326165 3034491071421 10246828768390 Table 4. The number of defining polynomials over F3 with λ ∈{1,2} m λ Number 1 1,2 2 2 1 4 2 2 3 3 1,2 6 4 1 14 4 2 11 5 1,2 26 6 1 68 6 2 63 7 1,2 158 8 1 424 8 2 411 9 1,2 1098 10 1 2980 10 2 2955 11 1,2 8054 12 1 22218 12 2 22151 is replaced with a new polynomial if this results in an increase in the minimum distance. This process is repeated until a code with the desired minimum distance is found or an iteration threshold is reached. The search is restarted periodically to ensure good coverage of the search space. It is not necessary to check the weight of every codeword in a QT code in order to determine the minimum distance d. Only a subset of the codewords need be considered since the Hamming weights of the polynomials rj(x) and ri(x) are the same if rj(x) = γx lri(x) mod (x m −λ), for integer l ≥ 0 and scalar γ ∈ Fq\{0}. 11 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Table 5. The number of defining polynomials over F4 with λ ∈{1,α,α2} m λ Number 1 1,α,α2 2 2 1,α,α2 4 3 1 10 3 α,α2 8 4 1,α,α2 24 5 1,α,α2 70 6 1 238 6 α,α2 232 7 1,α,α2 782 8 1,α,α2 2744 9 1 9726 9 α,α2 9710 10 1,α,α2 34990 11 1,α,α2 127102 12 1 466198 12 α,α2 466152 Table 6. The number of defining polynomials over F17 m λ Number 1 all 2 2 1,2,4,8,9,13,15,16 11 2 3,5,6,7,10,11,12,14 10 3 all 104 4 1,4,13,16 1317 4 2,8,9,15 1315 4 3,5,6,7,10,11,12,14 1306 5 all 17750 6 1,2,4,8,9,13,15,16 251543 6 3,5,6,7,10,11,12,14 251440 The best QT codes found over F17 are given in Tables 8 to 10, and over F19 in Tables 11 to 13. The defining polynomials are listed with the lowest degree coefficient on the left, i.e. 7321 corresponds to the polynomial x3 + 2x2 + 3x+ 7, with leading zeroes left out for brevity. The digits 10, 11,. . . , 18 are denoted by (10), (11), . . . , (18), respectively. As an example, consider the [24,4] code in Table 12 with m = 4, λ = 1 and p = 6 defining polynomials. These polynomials give the following generator matrix G =   1129 1596 1632 01(14)(11) 0016 169(12) 9112 6159 2163 (11)01(14) 6001 (12)169 2911 9615 3216 (14)(11)01 1600 9(12)16 1291 5961 6321 1(14)(11)0 0160 69(12)1   with weight distribution 12 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Table 7. The number of defining polynomials over F19 m λ number 1 all 2 2 1,4,5,6,7,9,11,16,17 12 2 2,3,8,10,12,13,14,15,18 11 3 1,7,8,11,12,18 130 3 2,3,4,5,6,9,10,13,14,15,16,17 128 4 1,4,5,6,7,9,11,16,17 1822 4 2,3,8,10,12,13,14,15,18 1811 5 all 27514 i Wi 0 1 20 6372 21 10944 22 28296 23 49680 24 35028 This code is optimal since it meets the Griesmer bound (1), and so establishes that d19(24,4) = 20. The codes that meet the Griesmer bound are indicated by ∗ in the tables. The codes given in the tables are QC codes (λ = 1) when a QC code has the highest minimum distance among all QT codes with the same length and dimension. In two cases for q = 19 and m = 4, a code with λ = −1 was found with a minimum distance higher than the corresponding QC code. 5. Conclusion Closed-form expressions for the number of twistulant matrices and corresponding defining polyno- mials were given. These polynomials were used in the construction of quasi-twisted codes, and several new optimal codes were obtained. Acknowledgment: The authors would like to thank Jumah Ali Algallaf for computing the numbers of defining polynomials. References [1] K. Betsumiya, S. Georgiou, T. A. Gulliver, M. Harada, C. Koukouvinos, On self-dual codes over some prime fields, Disc. Math. 262(1–3) (2003) 37–58. [2] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24(3-4) (1997) 235–265. [3] E. Z. Chen, N. Aydin, New quasi-twisted codes over F11–minimum distance bounds and a new database, J. Inform. Optimization Sci., 36(1-2) (2015) 129–157. 13 https://doi.org/10.1016/S0012-365X(02)00520-4 https://doi.org/10.1016/S0012-365X(02)00520-4 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1080/02522667.2014.961788 https://doi.org/10.1080/02522667.2014.961788 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 [4] E. Z. Chen, N. Aydin, A database of linear codes over F13 with minimum distance bounds and new quasi-twisted codes from a heuristic search algorithm, J. Algebra Comb. Discrete Appl., 2(1) (2015) 1–16. [5] J. A. Gallian, Contemporary Abstract Algebra, Eighth Edition, Brooks/Cole, Boston, MA 2013. [6] M. Grassl, Code Tables: Bounds on the parameters of various types of codes, available online at http://www.codetables.de. [7] P.P. Greenough, R. Hill, Optimal ternary quasi-cyclic codes, Des. Codes, Cryptogr. 2(1) (1992) 81–91. [8] T. A. Gulliver, Quasi-twisted codes over F11, Ars Combinatoria 99 (2011) 3–17. [9] T. A. Gulliver, New optimal ternary linear codes, IEEE Trans. Inform. Theory 41(4) (1995), 1182–1185. [10] T. A. Gulliver, V. K. Bhargava, Some best rate 1/p and rate (p−1)/p systematic quasi-cyclic codes over GF(3) and GF(4), IEEE Trans. Inform. Theory 38(4) (1992) 1369–1374. [11] T. A. Gulliver, V. K. Bhargava, New good rate (m − 1)/pm ternary and quaternary quasi-cyclic codes, Des. Codes, Cryptogr. 7(3) (1996) 223–233. [12] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, New York, NY 1977. [13] D. W. Newhart, On minimum weight codewords in QR codes, J. Combin. Theory Ser. A 48(1) (1988) 104–119. [14] V. Ch. Venkaiah, T. A. Gulliver, Quasi-cyclic codes over F13 and enumeration of defining polyno- mials, J. Discrete Algorithms 16 (2012) 249–257. 14 https://doi.org/10.13069/jacodesmath.36947 https://doi.org/10.13069/jacodesmath.36947 https://doi.org/10.13069/jacodesmath.36947 http://www.codetables.de http://www.codetables.de http://www.codetables.de https://doi.org/10.1007/BF00124211 https://doi.org/10.1007/BF00124211 https://doi.org/10.1109/18.391267 https://doi.org/10.1109/18.391267 https://doi.org/10.1109/18.144719 https://doi.org/10.1109/18.144719 https://doi.org/10.1023/A:1018090707115 https://doi.org/10.1023/A:1018090707115 https://doi.org/10.1016/0097-3165(88)90078-7 https://doi.org/10.1016/0097-3165(88)90078-7 https://doi.org/10.1016/j.jda.2012.04.006 https://doi.org/10.1016/j.jda.2012.04.006 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Appendix A: Table 8. QT codes over F17 with m = 3 code λ d bi(x) [6,3] 1 4∗ 11, 1(13)8 [9,3] 1 7∗ 18(11), 1, 117 [12,3] 1 10∗ 14(15), 1(12), 16, 157 [15,3] 1 13∗ 1, 118, 125, 152, 164 [18,3] 1 16∗ 118, 125, 1, 152, 13(12), 164 [21,3] 1 18∗ 152, 14(10), 13(16), 17(12), 127, 172, 115 [24,3] 1 21∗ 11, 137, 11(10), 145, 17(16), 18(11), 15(16), 13 [27,3] 1 24∗ 116, 12(11), 12(16), 18(11), 118, 11(13), 12, 147, 16(10) [30,3] 1 26 147, 11(14), 12(10), 16, 11, 164, 164, 1(13)8, 11(12), 182 [33,3] 1 29 184, 17(12), 14, 114, 17(10), 11(10), 19(16), 18, 12(13), 11(12), 1(13)4 [36,3] 1 32 18, 1(13)4, 12, 1(14), 1(12)8, 11(13), 14(13), 157, 18(12), 198, 1(10), 16(11) [39,3] 1 35∗ 116, 124, 117, 1(13)8, 11(11), 11(10), 12, 145, 19, 12(14), 1(12), 1(10)4, 15(14) [42,3] 1 38∗ 1(13), 15(12), 1, 152, 11, 125, 1(13)4, 12(11), 116, 175, 118, 198, 132, 137 [45,3] 1 40 1, 1(15), 118, 113, 1(12)8, 14(11), 124, 114, 19(16), 17, 126, 125, 154, 14, 11(16) [48,3] 1 43 15, 126, 14(15), 1(10)4, 15(16), 17(16), 11(14), 125, 12(12), 13(14), 143, 11(10), 1(10)(11), 11(16), 16(15), 157 [51,3] 1 46 18, 126, 198, 11(13), 157, 13(14), 15(14), 13(15), 12(16), 12(14), 12, 152, 13(10), 11, 16(11), 16(14), 14(11) [54,3] 1 49 11, 11(13), 154, 1, 135, 12, 152, 124, 143, 137, 14(11), 116, 16(14), 16(15), 13(12), 11(12), 1(10)8, 1(10) [57,3] 1 52∗ 11, 13(16), 1(13)4, 11(15), 1(13)8, 12, 15(14), 16(16), 12(11), 11(12), 18(12), 14(14), 1, 15(16), 125, 115, 147, 145, 162 [60,3] 1 54 12, 1, 113, 152, 12(10), 11(14), 114, 16(11), 15(12), 135, 1(10)(11), 17, 16, 126, 11, 134, 16(14), 157, 125, 165 [63,3] 1 57 11(13), 11, 12(15), 154, 19(16), 12, 12(10), 114, 125, 12(12), 15(14), 15(12), 117, 113, 14(10), 11(15), 16(15), 162, 1(13), 13(16), 123 [66,3] 1 60 13, 145, 15(12), 13(13), 117, 1(15), 12(10), 13(10), 143, 116, 14(13), 168, 16(11), 11, 11(12), 13(15), 1(14), 13(14), 115, 182, 17, 142 [69,3] 1 63 11(13), 1(12), 154, 1, 19, 128, 12(10), 182, 12(16), 12(15), 15(14), 18(12), 112, 11(12), 117, 198, 15(11), 1(11), 13(16), 11(14), 138, 1(13), 145 [72,3] 1 65 15, 175, 11, 13(12), 1(10), 11(12), 118, 19(14), 17(12), 16(11), 182, 154, 137, 138, 132, 16(15), 145, 18, 11(15), 125, 142, 147, 116, 17(10) [75,3] 1 68 1(10), 12, 12(13), 13(15), 126, 1(11), 12(16), 112, 11(13), 117, 12(12), 16(14), 14(13), 175, 1(10)4, 11, 14(11), 168, 11(16), 17(10), 17(16), 1(12), 1(12)8, 114, 127 15 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Table 9. QT codes over F17 with m = 4 code λ d bi(x) [8,4] 1 5∗ 1, 1128 [12,4] 1 9∗ 16(11)(14), 123, 1685 [16,4] 1 13∗ 12(10)(16), 1275, 113, 11(16)7 [20,4] 1 16 18, 13(11)7, 113(12), 1179, 12(15)2 [24,4] 1 20∗ 134, 1798, 18(11), 1(11)1(14), 126(10), 1(10)(11)(12) [28,4] 1 23 1(11), 1(13)1, 11(10)6, 137(15), 1(15)(11), 1(11)1(16), 1(10)5 [32,4] 1 27 1, 124, 13(14)7, 1(10)(11)(12), 126(16), 1(15)(16), 1437, 13(13)2 [36,4] 1 30 13, 15(16)(11), 13(15)(14), 1(10)2, 16(16), 11, 112(14), 12(13)(16), 1734 [40,4] 1 34 14(10)7, 1453, 1317, 1, 1(13)1, 11(15)4, 1598, 12(15)4, 1(12)2, 118(10) [44,4] 1 38 114(15), 12(10)(16), 11, 118(14), 113, 17(13)(16), 1(16)7, 14(15)(10), 1(12)7, 19(14), 147(16) [48,4] 1 41 12(12), 12, 11(16)(14), 181, 14, 111(14), 1254, 12(10)(13), 1114, 1(10)5, 156(10), 147(12) [52,4] 1 45 124(16), 14(13)3, 1114, 16, 1(15)(16), 12, 183(10), 118(16), 1192, 13(10), 12(14), 173(11), 14(15)(10) 116(15), 14(10)2, 1(11)64, 14(14)7 [60,4] 1 52 11, 11(13), 13(16), 146(13), 153, 13, 192, 164(10), 1386, 137(13), 1372, 125(16), 11(10)5, 1272, 12(13)3 [64,4] 1 56 13, 158(14), 11(15)(14), 1982, 1(13)5, 1(10)1(16), 12(14)5, 1517, 13(13)(12), 103, 11(12)9, 1(12)(15)8, 116(16), 11(15)7, 1239, 1(12)(10) [68,4] 1 59 12, 1(10)1(12), 12(13)(12), 1(11)2, 151(11), 17(12)(15), 16(11)(12), 1(15)(16), 1187, 1472, 124(15), 13(10)(13), 11(14), 1462, 1(11)3, 172(14), 11 [72,4] 1 63 1(15), 12(10)2, 11(16)(11), 11(16)7, 1(13), 15(16)5, 1838, 12(15), 14(15), 13(13)(11), 1682, 1(12)38, 1468, 135(15), 1795, 137(10), 14(10), 175 [76,4] 1 67 1(10), 12(16)7, 15(12)8, 1127, 13(15)5, 12(15)6, 1358, 126(16), 1123, 121(13), 14, 1387, 187, 1654, 11(12)(14), 1193, 163(11), 1(12)(13), 1468 [80,4] 1 70 11(13), 1, 13, 1(10)1(11), 15(12)4, 1268, 167, 11(12), 141(10), 184(15), 1517, 17(12)(11), 16(15)(14), 12(14)(15), 18(14), 1188, 17(10)(14), 138(12), 111(16), 176(16) [84,4] 1 74 1418, 17, 1356, 135(15), 13(16)2, 12(11)(13), 136, 1162, 16(10)(15), 149(16), 147(13), 12(16)(13), 17(16), 11(14)5, 1164, 1213, 121(15), 1838, 11(11)5, 1547, 17(10)4 [88,4] 1 78 1(10), 12(11)(16), 1(11)(13), 15(14), 13(16)8, 138(11), 133, 172(15), 1568, 13(16)(10), 132(14), 1(13)(15), 1169, 1(11)1, 17(15), 1146, 14(13)2, 14(10)(11), 11(10)(11), 14(11)(15), 14(13)(14), 15(11)(12) [92,4] 1 81 13, 11(16)2, 17(12)5, 16(15)(12), 11(10)(11), 1(13)(16), 1(10)(11), 119(11), 16(14), 163(11), 139, 1(12)2, 13(13)8, 15(16), 162, 128(13), 1652, 14(12), 16(16)5, 162(14), 118(15), 11(10)7, 1(10)1 [96,4] 1 85 1, 16(11)(14), 1145, 16(10)(14), 102, 14(11)(12), 1(10)8(11), 111, 11(11)2, 13(13)8, 176(10), 16(11)8, 1376, 12(11)5, 1(10)8(12), 187, 15(12)3, 1(12)3(11), 11(12)4, 17(12)(10), 111(12), 1(10)(12), 125(14), 1832 [100,4] 1 89 105, 1(12)3, 1235, 198(12), 13(15)3, 151(15), 134(10), 11, 146(10), 1(13)(15), 1(11)4, 11(12)(11), 184(15), 143(15), 113(14), 1158, 17(12)(10), 1173, 116(10), 183(10), 119(11), 19(15)4, 11(15)(13), 1584, 1425 16 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Table 10. QT codes over F17 with m = 5 code λ d bi(x) [10,5] 1 6∗ 18, 16(15)(16)2 [15,5] 1 10 11, 162(15)(14), 11(11)(15)(12) [20,5] 1 14 131, 11(13)96, 1041, 11(11) [25,5] 1 19 15, 11(11)(16)(12), 135(12)4, 1126(10), 1278(14) [30,5] 1 23 11(13)(10)(15), 1(11)(10)(16), 115, 115(11)4, 112(11)8, 17(12)8(11) [35,5] 1 28 111(16)6, 143(13)(16), 121, 106(13), 15(14)68, 119(16)6, 13(13)(16)7 [40,5] 1 32 19, 13415, 1576(16), 1547, 17(13)(15)(14), 12(12)(13)(12), 13(12)85, 12(13)(11)4 [45,5] 1 36 1(16)58, 131, 13(12)9, 1(15)7(16), 11(13)(15)(14), 12(15)3(15), 1(10)1(10), 13(14)3(12), 141(15)(12) [50,5] 1 41 16, 121(15)(13), 15217, 172(13)(12), 11132, 17(10)1(16), 13(11)75, 14(14)2(15), 12(11)26, 11(15)7(14) [55,5] 1 45 1, 12, 11(16)(14)(15), 1(15)(11)4, 1416(10), 11636, 19(10)7, 111(15)8, 11(13)7(16), 12495, 1748 [60,5] 1 50 19, 10(11)4, 1(10)83(11), 1(14)74, 11(13)(16)3, 11452, 13(10)(15)(14), 13(12)(14)(16), 12(12)34, 12(13)8(15), 17(10), 154(15)3 [65,5] 1 54 13, 11, 1058, 14(14)4, 126(14)8, 13(11)(12)(10), 121(11)8, 1389(12), 19(15)3, 129(12), 13(14)94, 12(15)27, 1212(13) [70,5] 1 59 11, 13475, 1(12)(10)(14), 11(13)4, 18(12)(15)2, 14(12)4, 14(13)(16)3, 173(11)(12), 11235, 1586(11), 1(16)6(12), 18(14)(10), 15(12)1(12), 1(14)9(10) [75,5] 1 63 1(10)(11), 1956, 19, 16(15)(12)(16), 19(16)4, 11(14)59, 112(10), 15(11)(12)8, 17(14)3, 17(16)42, 11(10)4(15), 11(11)3, 17(16)1(14), 12(15)1, 19(14)8 [80,5] 1 68 1, 1(16)(14)6, 14, 13(14)6(15), 163(11)(12), 111, 17595, 179(15)(14), 14382, 1(12)5(12)(10), 1(13)(14)9, 1(12)(15)6(16), 14(10)2, 12(14)15, 19(13)82, 1825(11) [85,5] 1 72 1(10), 1327(16), 11, 1(11)11, 177, 15(10)4, 11513, 13263, 155(14), 12(16)8(14), 1342(11), 123(15)9, 125(13)(14), 12(14)38, 1(12)5(12), 17(10)4(10), 1641(14) [90,5] 1 77 1(11), 119(13), 1933, 18198, 13714, 115(15)(15), 195(13), 17(11)3, 11475, 13(15)7, 156(10)2, 1(11)(16)(10), 15(16)2(15), 15(11)34, 113, 11(10)(12)3, 1(10)5(13), 11(14)(10)3 [95,5] 1 81 1, 12432, 1438(10), 13(11)1(12), 1(11)42, 131(10)(11), 14(11)1(15), 1425(11), 10(10), 1185(13), 1(12)18, 15982, 118(15)3, 1794(14), 12927, 1321(14), 13(14)(16)(14), 12(14)(12)(15), 1976(16) [100,5] 1 86 16(14)(13)(11), 15(14)72, 11, 1, 1431(11), 12(10)(13)3, 113(12)(14), 1156(10), 115(11), 1752(16), 12(12)(10)(13), 1247, 173(13)4, 172(15)(16), 1(13)(11)(11), 116(11)(14), 146(12), 16(15)(16)8, 11(11)(15)3, 157(16)4 [105,5] 1 90 13(13)(10)5, 1(11)37, 11, 1(14)(11)3, 117(14)3, 13(11)(16)(12), 13, 123(16)7, 1195, 19(12)9, 15(16)(12), 11(14)39, 11(14)6(14), 1(12)8, 15932, 1517(10), 1(12)(11)9, 156(10)(12), 13(10)6(13), 1183(13), 11(10)(16) [110,5] 1 95 12, 14, 13(12)9, 1(15)7(16), 11(13)(15)(14), 12(15)3(15), 1(10)1(10), 13(14)3(12), 141(15)(12), 15168, 11534, 1568(14), 118(15)(12), 1(10)44, 135(11)3, 163(11)(12), 11(15)(10)(16), 13(11)(15)(14), 169(16)(10), 117(11), 12(11)(14)(10), 14234 [115,5] 1 99 18, 1, 11(10)3(16), 1(11)(12)(15)8, 13(16)9(11), 13(16)95, 13(16)54, 1012, 1127(14), 15942, 131(16)2, 1187, 12(12)9(12), 1326, 15(14)1(14), 11112, 18(11)(12)8, 15(11)32, 164(10)8, 159(12)5, 1(14)(16)(13), 103(13), 17(11)4 [120,5] 1 104 14, 117(15)8, 1071, 15(13)(11)(15), 15(12)1, 13(12)(16)(13), 14(15)62, 134(10)(13), 15(16)85, 15(14)(16), 12, 19(11)(12), 1815, 112(14)(13), 14(11)2, 15(14)(10)8, 12(12)(11)(15), 1256(13), 1(12)3(12), 1452(16), 1243(10), 1166(14), 13(11)(10)4, 11(10)5(15) [125,5] 1 109 1(10), 13184, 13(13)(16), 12(13)53, 1081, 1(11)(10)(10), 13(11)2, 15(14)(15)2, 11(14)(11)9, 127(16)(13), 1415(12), 12972, 15175, 11(11)(12)6, 1(13)(16)(13), 19832, 14295, 13(16)(11), 12(16)4(15), 132(14)(15), 156(14)2, 11(12)87, 131(11)4, 15(16)64, 1656 17 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Table 11. QT codes over F19 with m = 3 code λ d bi(x) [6,3] 1 4∗ 1, 112 [9,3] 1 7∗ 135, 11, 16(16) [12,3] 1 10∗ 1, 15(12), 17(15), 138 [15,3] 1 13∗ 1(11), 154, 11(16), 149, 14(17 [18,3] 1 16∗ 17, 159, 11(16), 157, 145, 176 [21,3] 1 18 1, 114, 142, 12, 137, 159, 16(15) [24,3] 1 21∗ 11, 15(12), 1(11)(18), 138, 15(17), 139, 1(10)6, 159 [27,3] 1 24∗ 12, 18(14), 116, 162, 189, 154, 14(14), 13(12), 11(14) [30,3] 1 27∗ 1(11), 1, 15(11), 1(10)7, 1(13)6, 135, 11(13), 1(10)(18), 15(16), 128 [33,3] 1 29 112, 115, 1(12)(14), 12, 11, 15(12), 138, 1(17), 1(16), 128, 192 [36,3] 1 32 14(15), 15, 125, 117, 12(13), 158, 18(17), 11(13), 13(12), 1, 17(15), 11(17) [39,3] 1 35 167, 16, 11(11), 1(14), 17(12), 14, 1(12)(14), 173, 17, 13(16), 146, 127, 152 [42,3] 1 38∗ 1(11), 11(17), 17(18), 114, 12, 12(12), 15(17), 1(13)6, 18(17), 162, 11(12), 19, 116 1(10)(18) [45,3] 1 41∗ 14, 1(14)6, 16(14), 17, 11(11), 162, 17(12), 187, 16, 1(16), 1(10)(15), 1(10)6, 11(12) 178, 169 [48,3] 1 43 15(16), 1, 168, 12(12), 19(14), 129, 1(16), 12, 1(12)9, 154, 135, 189, 112, 1(10)9 1(10)8, 117 [51,3] 1 46 12, 18(17), 162, 15(11), 11(11), 137, 13(14), 15, 17(17), 11, 163, 1, 11(13), 12(15) 19(14), 135, 18(15) [54,3] 1 49 1, 18, 176, 13(16), 162, 12(12), 1(10), 125, 154, 192, 147, 1(10)6, 135, 159, 134, 169 13(12), 17 [57,3] 1 52 14, 13, 1, 1(13)6, 129, 143, 1(10), 168, 118, 158, 124, 1(10)(18), 18(14), 19(14), 14(15) 1(11)(18), 159, 12(13), 14(14) [60,3] 1 55 1(14), 1(10), 14, 1(17), 115, 126, 17(18), 154, 17, 1(13)6, 129, 12(15), 137, 11(16), 128 1(11)(18), 1(13)(18), 127, 113, 149 [63,3] 1 57 17, 1(12)9, 1(16), 1(14), 193, 135, 1(10)6, 12(16), 19, 1(18), 162, 116, 145, 126, 129 169, 14(17), 15(12), 11(16), 1(12), 1(14)6 [66,3] 1 60 1(12), 11(15), 15(18), 176, 137, 193, 138, 12(11), 1(10)9, 1(11)(18), 13(16), 135, 162 11(13), 16(15), 1(16)6, 17(18), 132, 17, 1(12)(14), 147, 125 [69,3] 1 63 1, 18, 1(16), 134, 1(13)(18), 14(15), 114, 1(14), 115, 11(11), 139, 186, 15(16), 12(15) 1(13), 15(17), 129, 1(10)8, 17(18), 14(11), 132, 14(14), 126 [72,3] 1 66 1(18), 12(15), 173, 117, 139, 1(15), 15(18), 1(12)9, 167, 1(13), 11, 127, 1(13)6, 11(16) 11(14), 13(16), 19, 168, 129, 13(17), 1(11)(18), 138, 14(11), 16(16) [75,3] 1 69 15, 12(14), 13(17), 149, 1, 1(14), 193, 11(13), 134, 16(16), 1(13)(18), 15(16), 115, 1(13) 14(15), 12(17), 1(11)(18), 128, 116, 16(14), 17(17), 15(12), 186, 137, 11 18 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Table 12. QT codes over F19 with m = 4 code λ d bi(x) [8,4] 1 5∗ 12, 114 [12,4] 1 9∗ 1(10), 1193, 1(10)6(15) [16,4] -1 13∗ 12(12), 16, 134(12) , 111(14) [20,4] -1 17∗ 14(17), 115(17), 1281, 11(11)9, 1146 [24,4] 1 20∗ 1129, 1596, 1632, 1(14)(11), 16, 169(12) [28,4] 1 23 18, 135(14), 19, 16(14)(17), 11(15), 145(11), 14(11)(15) [32,4] 1 27 104, 112, 1534, 1629, 1737, 17(15)6, 12(16)6, 1356 [36,4] 1 30 18, 119(12), 12(16), 112, 15(11)(15), 13(16)3, 121(18), 15(18)(11), 1719 [40,4] 1 34 106, 14(14)3, 1627, 11(12)(18), 16(10)(14), 14(11)(12), 157(15), 152(16), 1(16)9 1923 [44,4] 1 38 16, 127(14), 12(14)(16), 15, 1(10)(18)7, 18(17)6, 149(15), 1398, 158, 1279, 141(18) [48,4] 1 41 159(14), 11, 15(17), 121(13), 15(13)(16), 14(10)(17), 12(14)6, 124(18), 146(11), 13(11) 1168, 118(15) [52,4] 1 45 1(12), 1(12)5, 14(11), 123(18), 14(10)(18), 13(15)(12), 12(14)(11), 15, 187(17) 163(17), 159(11), 17(12), 1174 [56,4] 1 49 13, 11(15)(13), 1(10)5(14), 1(11)59, 161(14), 1619, 13(10)(15), 1(12)(16), 11(12)(11) 1(10)(12), 12(15)(18), 13(16)6, 126(11), 1272 [60,4] 1 52 1843, 169, 17(17), 1153, 11(11)2, 1(11)8(14), 183(17), 16(16)2, 141(12), 12(14)(15) 11(15), 1416, 1576, 148, 162(17) [64,4] 1 56 1(14), 153(15), 169(16), 11(12)(16), 1157, 19, 1(12)4, 1427, 124(11), 1144, 12(16)2 1(16)7, 14(17)9, 12(11)7, 169(15), 1188 [68,4] 1 60 14, 1283, 1642, 15(17)(16), 1(16)(10), 1(11)7, 1(12)29, 12, 1923, 1287, 17(15)8, 113 12(14)3, 131(18), 17(10), 1582, 149(12) [72,4] 1 63 14, 15(14)6, 1(11)9, 1215, 19(15), 15(17)(18), 1, 168(18), 1278, 11(17)4, 13(12)8 1(11)7(15), 16(14)(15), 1192, 164, 1(11)(16)6, 111(12), 132 [76,4] 1 67 1(10), 152(17), 1(12)(13), 113(17), 1(11)4, 12(13)(14), 1(11)78, 191(14), 191, 13(12)3 14(12), 1358, 1696, 11(18)(15), 1(12)2(15), 1(12)8, 1(18)7, 1194, 1(13)(13) [80,4] 1 71 15, 14(11)4, 12(18)2, 1(15)9, 1(13)6(14), 129(14), 11(13)(16), 1153, 12(14)(13), 128 1(16)(17), 167(12), 12(10)(16), 13, 11(11)(15), 14(14)(15), 12(17), 117(10), 1(12)2(15) 14(18)(12) [88,4] 1 78 1(11), 1629, 11(11)(15), 178(12), 14(17)9, 113(11), 12(18)(16), 12(13)(15), 1923 113(12), 11(17), 14(15)4, 1287, 125(13), 14(10)5, 102, 1(12)78, 14(14)(17), 1(11)53 19(17), 154(17), 12(10)(12) [92,4] 1 82 1(12), 13(18)(14), 13(13)6, 1319, 115(15), 106, 164, 12(17)7, 19(12)7, 14(14)3, 13(13)3 14(17)9, 19(12)(18), 112(17), 138(17), 1(11)(17), 1175, 15(13)(12), 168, 1(10)58, 181 16(13)3, 178(12) [96,4] 1 86 17, 1(11)(10)8, 18(17)(14), 1(18)4, 145(17), 13(17)(18), 1(11)(15), 12(14)(15), 1293 11(16)9, 164(14), 1532, 1(10)78, 12(10)(11), 14(13)4, 14(14)2, 186(14), 111(16) 15(13)9, 1569, 132(17), 13, 11(16)(14), 12(11) [100,4] 1 89 15, 1, 1(13)9, 131(15), 14(10)7, 1(10)5(18), 11(16)7, 159(16), 16(14), 138(12), 11(18)6 11(11)(13), 118(10), 12(12)7, 115, 19(11)2, 11(14)(15), 13, 12(17)4, 1(10)(18), 17(18)3 17(10)(17), 15(12)8, 1(11)78, 141(12) 19 T. A. Gulliver, V. Ch. Venkaiah / J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–18 Table 13. QT codes over F19 with m = 5 code λ d bi(x) [10,5] 1 6∗ 104, 1(10)(18)59 [15,5] 1 11∗ 114, 11(15)(13)(10), 14(14)(10)3 [20,5] 1 15 105, 11(15)(18)(12), 12(18)(15)(18), 131(12)6 [25,5] 1 19 12, 117(10)(15), 198, 12(12)(11)(13), 1(15)(14)4 [30,5] 1 23 11, 11(15)8(15), 11(16)(13)9, 1(11)12, 11634, 1(16)(18)8 [35,5] 1 28 132(14)(15), 11, 10(13)(18), 11(13)(11)5, 135(12)(15), 159(12), 1(14)(18)1 [40,5] 1 32 1(13)1, 1185(15), 17637, 16(13)(11)8, 1(15)(13)2, 131(14)(11), 17(12)1(14) 1(18)(13)(14) [45,5] 1 37 1(12), 1(16)(10)7, 13(11)(15)4, 17(10)53, 1567(18), 12(17)62, 1(11)71(15) 16(10)8(17), 119(16)3 [50,5] 1 41 1(15)(15)6, 1298(11), 13(18)4(14), 13, 143(18)9, 1(14)(13)(11), 11766, 1(13)9(12) 124(10)(16), 19(13)(11)2 [55,5] 1 46 1(16), 14(13)(17)(12), 143(18)(11), 11(12)7(10), 13(17)(16)(11), 12347, 1(10)(15)93 138(12)7, 138(14)(17), 11(13)(11)6, 1(15)(11) [60,5] 1 50 1, 15(12)4(18), 11345, 1562(15), 12145, 12(10)5(17), 108, 18(16)5, 13(18)8(12) 12(16)(12)(16), 15(18)(13)(15), 12634 [65,5] 1 55 11(10), 11(12)8(10), 16843, 137(13)4, 18(14)(16)6, 1(10)2, 13(11)(10)5, 1237(15) 1287, 17(18)57, 13(17)95, 13(15)(18)8, 13235 [70,5] 1 59 13, 13(11)(18)8, 1146(13), 11524, 13739, 12(10)(13)(16), 108(10), 16953, 17(11)(17) 13(17)74, 1535, 18387, 16(13)(12)(11), 141(15)(12) [75,5] 1 64 13, 1(13)(11)(13)(18), 12, 13(16)(18)4, 189(13)(17), 12(18)(15), 1177(17), 145(13) 12(13)(12)2, 15(17)(16)6, 1013, 11(17)(15)(11), 117(15), 11112, 14(14)57 [80,5] 1 68 128(17)3, 1934(18), 18, 132(10)(18), 184(14)6, 1(16)52, 133(16), 13(13)(15)2 12(15)43, 13(16)17, 14(14)15, 17586, 12(15)15, 11962, 17(14), 117(12)9 [85,5] 1 73 1(15), 13(16)2, 143(11)9, 12(16)(17)4, 1136(11), 13(18)46, 15(17)59, 1163(17), 1 126(15)6, 13(16)56, 161(12)8, 10(16), 15(18)5(18), 12(15)(16)(18), 135(14)(11) 17(17)(15) [90,5] 1 77 101, 102, 11(11)(11)(18), 1418(17), 1274, 12(16)7, 13(17)(12)(18), 11(13)(16)(11) 1693, 13(18)(14)(12), 15(17)1(11), 11778, 1(15)1, 13854, 1457, 11(14)(17)4, 1135 12(13)(12)(14) [95,5] 1 82 1, 12(14)3(17), 101, 11(17)7(12), 14(17)(14)(15), 137(17)(16), 12(10)49 1(13)(14)(16), 12(14)68, 114, 12(16)1(12), 134(16)5, 18(12)4, 18(13)2, 161(15)2 1(10)(18)(11), 11(14)4(14), 112(15)(14), 12(12)43 [100,5] 1 87 111, 14, 183(16), 192(13)(17), 17(17)(14)6, 11(15)38, 13(13)(18)(15), 13(18)7(12) 1(12)(16)1, 13(13)(15)5, 1354(14), 164(18)(11), 1585, 145(11)5, 1(11)(17)4 1(10)(18)(17), 11(15)(17)(11), 17(18)7(18), 117(18)(12), 13(14)46 [105,5] 1 91 10(14), 11, 14(12)(15), 1118(17), 12, 1(13)(14)3, 18(14)6(15), 14(13)46, 137(13)9 11(16)(17)6, 1247(12), 1354(16), 11(10)27, 115(14)8, 131(12)3, 12(18)78, 114(14)2 11(11)75, 1538, 1(16)(16)2, 16(11)4 [110,5] 1 96 1025, 14, 18, 193(10)(17), 12(10)4(11), 1(10)8(17)(14), 14(13)2(18), 1(18)(18)(16) 137(10)(14), 11(18)78, 158(11)(15), 111(17)6, 1(13)(11)(13)6, 12(18)42 13(12)(16)(12), 11484, 1383(15), 142(12)6, 12(15)2(17), 11279, 14(11)(16)(17) 1693(17) [115,5] 1 100 12, 1(14)5, 1(11), 145(16)5, 131(14)(18), 11(16)(17)(10), 177(12), 198(13), 161(14)6 14(13)(10)(14), 181(13), 1239, 11(13)9(14), 18(11)7(15), 12(10)1(16), 13(12)3(15) 167(12)(18), 11277, 19(10)1, 1442, 115(13)(17), 1(17)(10)4, 15(11)(12)6 [120,5] 1 105 118, 1769(12), 1, 142(10)3, 1(14)(17)(14), 12(12)(17)(18), 14(10)8(17), 1616 1529(16), 1(10)(17)4(14), 1144(16), 135(12), 15(13)4(14), 11853, 1(12)(10)(17) 11(10), 153(13)8, 13(14)34, 18(14)(10)(14), 13(11)1(18), 12(17)5(14), 11(15)(17)(16) 1(17)42, 123(17)(16) [125,5] 1 109 102, 17(16)7, 13894, 13(15)(18)(14), 13(18)(10)(15), 12(14)76, 11, 15696, 142(13)3 11(11)(12)9, 15376, 11(13)76, 115(10), 1(10)52, 112(17)(10), 1145(10), 18957 132(11)2, 132(10)6, 118(11)5, 12(16), 13(14)2(17), 13(13)(10)3, 15(18)32 134(11)(15) 20 Introduction Quasi-twisted codes Defining polynomials Quasi-twisted codes over F17 and F19 Conclusion References