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In this paper, we solve the reversibility problem for DNA codes over the non-chain ring $R_{k,s}=\mathbb{F}_{4^{2k}}[u_1,...,u_{s}]/< u_1^2-u_1,..., u_s^2-u_s>$. We define an automorphism $\theta$ over $R_{k,s}$ which help us both find... more
In this paper, we solve the reversibility problem for DNA codes over the non-chain ring $R_{k,s}=\mathbb{F}_{4^{2k}}[u_1,...,u_{s}]/< u_1^2-u_1,..., u_s^2-u_s>$. We define an automorphism $\theta$ over $R_{k,s}$ which help us both find the idempotent decomposition of $R_{k,s}$ and solve the reversibility problem via skew cyclic codes. Moreover, we introduce a generalized Gray map that preserves DNA reversibility.
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In this work, we focus on cyclic codes over the ring F2+uF2+vF2+uvF2, which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (F2 + uF2 +... more
In this work, we focus on cyclic codes over the ring F2+uF2+vF2+uvF2, which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (F2 + uF2 + vF2 + uvF2)/(x − 1) and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over F2+uF2+vF2+uvF2 under two Gray maps that are defined. We also characterize the binary images of cyclic codes over F2 + uF2 + vF2 + uvF2 in general.
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In this work, we investigate linear codes over the ring [FORMULA] . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary... more
In this work, we investigate linear codes over the ring [FORMULA] . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43-65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32-45, 1999). We then characterize the [FORMULA] -linearity of binary codes under the Gray map and give a main class of binary codes as an example of [FORMULA] -linear codes. The duals and the complete weight enumerators for [FORMULA] -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over [FORMULA] are obtained.
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In this work codes over one of seven local Frobenius non-chain rings of order 16 are studied. The ring structure is discussed showing both the similarities and differences to a previously studied ring. A duality preserving Gray map is... more
In this work codes over one of seven local Frobenius non-chain rings of order 16 are studied. The ring structure is discussed showing both the similarities and differences to a previously studied ring. A duality preserving Gray map is given and is used to present MacWilliams identities and self-dual codes. Connections between these self-dual codes and real unimodular lattices are also discussed. Some extremal Type II Z
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We describe skew G-codes, which are codes that are the ideals in a skew group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. These codes generalize many of the well-known classes of codes... more
We describe skew G-codes, which are codes that are the ideals in a skew group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. These codes generalize many of the well-known classes of codes such as cyclic, quasicyclic, constacyclic codes, skew cyclic, skew quasicyclic and skew constacyclic codes. Additionally, using the skew G-matrices, we can generalize almost all the known constructions in the literature for self-dual codes.
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ABSTRACT In this paper skew cyclic codes over the the family of rings Fq+vFq with v2 = v are studied for the �rst time in its generality. Structural properties of skew cyclic codes over Fq + vFq are investigated through a decomposition... more
ABSTRACT In this paper skew cyclic codes over the the family of rings Fq+vFq with v2 = v are studied for the �rst time in its generality. Structural properties of skew cyclic codes over Fq + vFq are investigated through a decomposition theorem. It is shown that skew cyclic codes over this ring are principally generated. The idempotent generators of skew-cyclic codes over Fq and Fq+vFq have been considered for the �rst time in literature. Moreover, a BCH type bound is presented for the parameters of these codes.
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In this work, linear codes over the ring S4 = F2 + uF2 + u2F2 + u3F2 are considered. The Lee weight and gray map for codes over S4 are defined and MacWilliams identities for the complete, the symmetrized and the Lee weight enumerators are... more
In this work, linear codes over the ring S4 = F2 + uF2 + u2F2 + u3F2 are considered. The Lee weight and gray map for codes over S4 are defined and MacWilliams identities for the complete, the symmetrized and the Lee weight enumerators are obtained. Cyclic and (1 + u2)-constacyclic codes over S4 are studied, as a result of which a substantial number of optimal binary codes of different lengths are obtained as the Gray images of cyclic and constacyclic codes over S4.
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The quasi-cyclic moderate-density parity-check code (QC-MDPC) cryptosystem is one of the recent variants of the original McEliece code-based cryptosystem, that has also been part of the BIKE cryptosystem, which has been submitted to NIST... more
The quasi-cyclic moderate-density parity-check code (QC-MDPC) cryptosystem is one of the recent variants of the original McEliece code-based cryptosystem, that has also been part of the BIKE cryptosystem, which has been submitted to NIST as a post-Quantum cryptosystem candidate. We show that in certain cases the secret key can be recovered from the public key by means of a polynomial factorization. This leads to the concept of "weak keys" for the cryptosystem. Even though the probability of choosing a weak key at random is low, we are able to find weak keys quite easily. This suggests that avoiding weak keys may be introduced as a condition in the implementation of the cryptosystem.
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In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets F_2, F_2+uF_2, F_4+uF_4, we were able to obtain extremal binary self-dual codes of lengths 40, 64 including... more
In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets F_2, F_2+uF_2, F_4+uF_4, we were able to obtain extremal binary self-dual codes of lengths 40, 64 including new extremal binary self-dual codes of length 68. More precisely, 43 new extremal binary self-dual codes of length 68, with rare new parameters have been constructed
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Dear Colleagues The purpose of this special issue of Journal of Algebra,Combinatorics, Discrete Structures and Applications was to collect a sample of papers in active areas of research in algebraic coding theory and its connections to... more
Dear Colleagues The purpose of this special issue of Journal of Algebra,Combinatorics, Discrete Structures and Applications was to collect a sample of papers in active areas of research in algebraic coding theory and its connections to other areas. A number of researchers submitted manuscripts to the special issue. After a thorough review process, six articles have been selected to appear in the special issue. We thank all researchers who submitted an article. Their contributions are sincerely appreciated, regardless of whether they have been accepted for publication or not. We are particularly grateful to our small number of dedicated reviewers who did a meticulous job of reviewing in a short period of time. The articles selected for this special issue are a representative sample of the current research trends in algebraic coding theory. In their article "Construction of quasi-twisted codes and enumeration of defining polynomials", Gulliver and Venkaiah enumerate all twis...
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We give an elementary approach to proving divisibility results for a class of binomial sums that are similar to Fleck's congruence. We use tools that are accessible to undergraduate students and in proving the divisibility results, we... more
We give an elementary approach to proving divisibility results for a class of binomial sums that are similar to Fleck's congruence. We use tools that are accessible to undergraduate students and in proving the divisibility results, we obtain additional interesting properties that we highlight in several parts of the paper.
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We introduce a new extension of the Lee weight to Ζpk and later to Galois rings GR(pk,m). The weight we define is a non-homogeneous weight and is different than the one that is generally labeled as "generalized Lee weight".... more
We introduce a new extension of the Lee weight to Ζpk and later to Galois rings GR(pk,m). The weight we define is a non-homogeneous weight and is different than the one that is generally labeled as "generalized Lee weight". Unlike the case of generalized Lee weight, we define a distance-preserving Gray map from (Ζpk, extended Lee distance)to (Fppk-1, Hamming distance), thus making our extension practical for coding theory purposes.
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The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the... more
The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enu...
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In this work, we focus on cyclic codes over the ring [FORMULA] , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring [FORMULA] and... more
In this work, we focus on cyclic codes over the ring [FORMULA] , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring [FORMULA] and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over [FORMULA] under two Gray maps that are defined. We also characterize the binary images of cyclic codes over [FORMULA] in general.
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In this work, linear codes over Z 2 s are considered together with the extended Lee weight that is defined as w L (x) = x if x ≤ 2 s−1 , 2 s − x if x &gt; 2 s−1. The ideas used by Wilson and Yıldız are employed to obtain di-visibility... more
In this work, linear codes over Z 2 s are considered together with the extended Lee weight that is defined as w L (x) = x if x ≤ 2 s−1 , 2 s − x if x &gt; 2 s−1. The ideas used by Wilson and Yıldız are employed to obtain di-visibility properties for sums involving binomial coefficients and the extended Lee weight. These results are then used to find bounds on the power of 2 that divides the number of codewords whose Lee weights fall in the same congruence class modulo 2 e. Comparisons are made with the results for the trivial code and the results for the homogeneous weight.
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Lattice and code cryptography can replace existing schemes such as elliptic curve cryptography because of their resistance to quantum computers. In support of public key infrastructures, the distribution, validation and storage of the... more
Lattice and code cryptography can replace existing schemes such as elliptic curve cryptography because of their resistance to quantum computers. In support of public key infrastructures, the distribution, validation and storage of the cryptographic keys is then more complex for handling longer keys. This paper describes practical ways to generate keys from physical unclonable functions, for both lattice and code-based cryptography. Handshakes between client devices containing the physical unclonable functions (PUFs) and a server are used to select sets of addressable positions in the PUFs, from which streams of bits called seeds are generated on demand. The public and private cryptographic key pairs are computed from these seeds together with additional streams of random numbers. The method allows the server to independently validate the public key generated by the PUF, and act as a certificate authority in the network. Technologies such as high performance computing, and graphic pr...
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Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family Rk, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We... more
Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family Rk, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of codes over Rk under this Gray map. We then discuss quasi-twisted codes over Rk and their binary images under the homogeneous Gray map. In this way, we find many optimal binary codes which are self-orthogonal and quasi-cyclic. In particular, we find a substantial number of optimal binary codes that are quasi-cyclic of index 8, 16 and 24, nearly all of which are new additions to the database of quasi-cyclic codes kept by Chen.
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Linear codes over the ring ℤ4 + uℤ4 have been introduced recently. In this paper, we study cyclic codes over this ring. We determine algebraic structures of cyclic codes over ℤ4 + uℤ4 and obtain basic facts about their generators. Making... more
Linear codes over the ring ℤ4 + uℤ4 have been introduced recently. In this paper, we study cyclic codes over this ring. We determine algebraic structures of cyclic codes over ℤ4 + uℤ4 and obtain basic facts about their generators. Making use of their algebraic structure, we conducted a computer search for cyclic codes over ℤ4 + uℤ4 and obtained many new linear codes over ℤ4. These codes have been added to the database of ℤ4 codes.
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2e is divisible by 2 ⌊ k−2e−2 2e−2 ⌋ . We prove this result by introducing a lemma and applying the lemma in one of the theorems proved by Wilson. The method used is different than the one used in our previous work on Lee weight... more
2e is divisible by 2 ⌊ k−2e−2 2e−2 ⌋ . We prove this result by introducing a lemma and applying the lemma in one of the theorems proved by Wilson. The method used is different than the one used in our previous work on Lee weight enumerators in which more general results were ...
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In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets F_2, F_2 + uF_2, F_4+uF_4, we were able to obtain extremal binary self-dual codes of lengths 40, 64... more
In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets F_2, F_2 + uF_2, F_4+uF_4, we were able to obtain extremal binary self-dual codes of lengths 40, 64 including new extremal binary self-dual codes of length 68. More precisely, 43 new extremal binary self-dual codes of length 68, with rare new parameters have been constructed.
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In this paper, we construct self-dual codes from a construction that involves both block circulant matrices and block quadratic residue circulant matrices. We provide conditions when this construction can yield self-dual codes. We... more
In this paper, we construct self-dual codes from a construction that involves both block circulant matrices and block quadratic residue circulant matrices. We provide conditions when this construction can yield self-dual codes. We construct self-dual codes of various lengths over F2 and F2 + uF2. Using extensions, neighbours and sequences of neighbours, we construct many new self-dual codes. In particular, we construct one new self-dual code of length 66 and 51 new self-dual codes of length 68.
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In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the... more
In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings \begin{document}$ \mathbb{F}_2+u \mathbb{F}_2 $\end{document} and \begin{document}$ \mathbb{F}_4+u \mathbb{F}_4 $\end{document} , we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally self-dual codes with parameters \begin{document}$ [72,36,14] $\end{document} .
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A binary linear [2n, n]-code with generator matrix [In|A] can be associated with a digraph on n vertices with adjacency matrix A and vice versa. We use this connection to present a graph theoretic formula for the minimum distance of codes... more
A binary linear [2n, n]-code with generator matrix [In|A] can be associated with a digraph on n vertices with adjacency matrix A and vice versa. We use this connection to present a graph theoretic formula for the minimum distance of codes with information rate 1/2. We also formulate the equivalence of such codes via new transformations on corresponding digraphs.
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In this work, extension theorems are generalized to self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over F2m + uF2m for m = 1, 2. The duality and distance preserving... more
In this work, extension theorems are generalized to self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over F2m + uF2m for m = 1, 2. The duality and distance preserving Gray maps from F4 + uF4 to (F2 + uF2) and F42 are used to obtain self-dual codes whose binary Gray images are [64, 32, 12]-extremal self-dual. An F2 + uF2-extension is used and as binary images, 178 extremal binary self-dual codes of length 68 with new weight enumerators are obtained. Especially the first examples of codes with γ = 3 and many codes with the rare γ = 4, 6 parameters are obtained. In addition to these, two hundred fifty doubly even self dual [96, 48, 16]-codes with new weight enumerators are obtained from four-circulant codes over F4 + uF4. New extremal doubly even binary codes of lengths 80 and 88 are also found by the F2+uF2-lifts of binary four circulant codes and a corresponding result about 3-designs is stated.
We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F_2 + uF_2 and F_4 + uF_4; using groups of order 3, 7, 9, 13, and 15.... more
We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F_2 + uF_2 and F_4 + uF_4; using groups of order 3, 7, 9, 13, and 15. Through these constructions and their extensions, we find binary self-dual codes of lengths 32, 40, 56, 64, 68 and 80, all of which are extremal or optimal. In particular, we find five new self-dual codes of parameters [56, 28, 10], twenty-three extremal binary self-dual codes of length 68 with new weight enumerators and fifteen new self-dual codes of parameters [80, 40, 14].
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In this work, construction methods for formally self-dual codes are generalized in the form of block lambda-circulant matrices. The constructions are applied over the rings F_2,R1 = F_2 + uF_2 and S = F_2[u]=(u^3-1). Using n-block... more
In this work, construction methods for formally self-dual codes are generalized in the form of block lambda-circulant matrices. The constructions are applied over the rings F_2,R1 = F_2 + uF_2 and S = F_2[u]=(u^3-1). Using n-block lambda-circulant matrices for suitable integers n and units lambda, many binary FSD codes (as Gray images) with a higher minimum distance than best known self-dual codes of lengths 34, 40, 44, 54, 58, 70, 72 and 74 were obtained. In particular, ten new even FSD [72, 36, 14] codes were constructed together with eight new near-extremal FSD even codes of length 44 and twentyfive new near-extremal FSD even codes of length 36.
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Binary linear codes are constructed from graphs, in particular, by the generator matrix [In|A] where A is the adjacency matrix of a graph on n vertices. A combinatorial interpretation of the minimum distance of such codes is given. We... more
Binary linear codes are constructed from graphs, in particular, by the generator matrix [In|A] where A is the adjacency matrix of a graph on n vertices. A combinatorial interpretation of the minimum distance of such codes is given. We also present graph theoretic conditions for such linear codes to be Type I and Type II self-dual. Several examples of binary linear codes produced by well-known graph classes are given.