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Hulls of linear codes are widely studied due to their good properties and wide applications. Let n = q m - 1 r and C be an [n, k] cyclic code over F q , where r | q - 1 . In this paper, we present several necessary and sufficient conditions for BCH codes of length n that have k - 1 or k ⊥ - 1 dimensional hulls, where k ⊥ is the dimension of C ⊥ . Further, we give the parameters of several families of self-orthogonal codes that arise as hulls of BCH codes. We obtain many optimal codes. [ABSTRACT FROM AUTHOR]

Recently, a new algorithm to test equivalence of two cyclic codes has been introduced which is efficient and produced useful results. In this work, we generalize this algorithm to constacyclic codes. As an application of the algorithm we found many constacyclic codes with good parameters and properties. In particular, we found 22 new codes that improve the minimum distances of best known linear codes (BKLCs). [ABSTRACT FROM AUTHOR]

In this paper, let α + u β be a unit in F p m + u F p m (u 2 = 0) , where p is an odd prime, m is a positive integer and β ≠ 0 . With the help of decomposition of the binomial x 8 - α 0 into a product of irreducible coprime polynomials and the ring (F p m + u F p m) [ x ] x 8 p s - (α + u β) is a principal ideal ring, we give the complete description of all (α + u β) -constacyclic codes of length 8 p s over the finite commutative chain ring F p m + u F p m (u 2 = 0) in terms of their generator polynomials, where α 0 p s = α . We also find out the number of codewords in each of these constacyclic codes. Besides illustrating our results with examples, we determine duals of constacyclic codes, and as an application, we determine the self-dual, self-orthogonal, dual-containing, and linear complimentary-dual (α + u β) -constacyclic codes of length 8 p s over F p m + u F p m (u 2 = 0) . Also, we determine the RT (Rosenbloom–Tsfasman) distances, RT weight distributions, and Hamming distances of such constacyclic codes. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, q = p m , R 1 = F q + u F q + v F q + u v F q with u 2 = 1 , v 2 = 1 , u v = v u and R 2 = F q [ u , v , w ] / ⟨ u 2 - 1 , v 2 - 1 , w 2 - 1 , u v - v u , v w - w v , w u - u w ⟩ with u 2 = 1 , v 2 = 1 , w 2 = 1 , u v = v u , v w = w v , w u = u w . In this paper, F q R 1 R 2 -cyclic codes are introduced. We construct quantum error-correcting codes from F q R 1 R 2 -cyclic codes and introduced a Gray map to find new and better quantum error-correcting codes than previously known quantum error-correcting codes over F q . [ABSTRACT FROM AUTHOR]

A new class of binary quantum codes from cyclic codes over Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ , u 4 = 0 , is introduced. The generator polynomials of Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ -cyclic codes of length (r, s) are obtained through the factorization of x r - 1 and x s - 1 into pairwise coprime monic polynomials over Z 2 , where r and s are odd positive integers. A minimal spanning set for these codes is obtained. Under some restricted conditions, the structure of the duals of Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ -cyclic codes is also determined. Necessary and sufficient conditions for a Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ -cyclic code of this restricted class to contain its dual or to be self-orthogonal are obtained. A new Gray map is defined, and the binary quantum codes are obtained by using the Calderbank-Shor-Steane construction on self-orthogonal or dual containing Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ -cyclic codes. Some examples of binary quantum codes with good parameters constructed from Z 2 Z 2 [ u ] / ⟨ u 4 ⟩ -cyclic codes are given. [ABSTRACT FROM AUTHOR]

In this paper, we study linear codes over ring Rk = 𝔽pm[u1, u2,⋯,uk]/〈ui2$\begin{array}{} u^{2}_{i} \end{array} $ = ui, uiuj = ujui〉 where k ≥ 1 and 1 ≤ i, j ≤ k. We define a Gray map from RkntoFpm2kn$\begin{array}{} R_{k}^n\,\,\text{to}\,\,{\mathbb F}_{p^m}^{2^kn} \end{array} $ and give the generator polynomials of constacyclic codes over Rk. We also study the MacWilliams identities of linear codes over Rk.

In this paper, quantum codes over Fp from cyclic codes over the ring Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉, where u2 = 1, v3 = v, uv = vu and p is an odd prime have been studied. We give the structure of cyclic codes over the ring Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉 and obtain quantum codes over Fp using self-orthogonal property of these classes of codes. Moreover, by using decomposing method, the parameters of the associated quantum code have been determined. [ABSTRACT FROM AUTHOR]

In this paper, we study quantum codes over $$F_q$$ from cyclic codes over $$F_q+uF_q+vF_q+uvF_q,$$ where $$u^2=u,~v^2=v,~uv=vu,~q=p^m$$ , and p is an odd prime. We give the structure of cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$ and obtain self-orthogonal codes over $$F_q$$ as Gray images of linear and cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$ . In particular, we decompose a cyclic code over $$F_q+uF_q+vF_q+uvF_q$$ into four cyclic codes over $$F_q$$ to determine the parameters of the corresponding quantum code. [ABSTRACT FROM AUTHOR]

Constacyclic codes form an interesting family of error-correcting codes due to their rich algebraic structure, and are generalizations of cyclic and negacyclic codes. In this paper, we classify repeated-root constacyclic codes of length ℓ p over the finite field $\mathbb {F}_{p^{m}}$ containing p elements, where ℓ ≡ 1(mod 2), p are distinct primes and t, s, m are positive integers. Based upon this classification, we explicitly determine the algebraic structure of all repeated-root constacyclic codes of length ℓ p over $\mathbb {F}_{p^{m}}$ and their dual codes in terms of generator polynomials. We also observe that self-dual cyclic (negacyclic) codes of length ℓ p over $\mathbb {F}_{p^{m}}$ exist only when p = 2 and list all self-dual cyclic (negacyclic) codes of length ℓ2 over $\mathbb {F}_{2^{m}}$. We also determine all self-orthogonal cyclic and negacyclic codes of length ℓ p over $\mathbb {F}_{p^{m}}$. To illustrate our results, we determine all constacyclic codes of length 175 over $\mathbb {F}_{5}$ and all constacyclic codes of lengths 147 and 3087 over $\mathbb {F}_{7}$. [ABSTRACT FROM AUTHOR]