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In this paper, we investigate the Gray images of codes over chain rings, leading to the derivation of infinite families of self-orthogonal linear codes over the residue field Fq. We determine the parameters of optimal self-orthogonal and divisible linear codes. Additionally, we study the Gray images of quasi-twisted codes, resulting in some self-orthogonal Griesmer quasi-cyclic codes. Finally, we employ the CSS construction to derive some quantum codes based on self-orthogonal linear codes. [ABSTRACT FROM AUTHOR]

In this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary self-dual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92. [ABSTRACT FROM AUTHOR]

Featured Application: Using physical unclonable functions (PUFs), in support of networks secured with a public key infrastructure, to generate, on demand, the key pairs needed for lattice and code PQC algorithms. 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 processing units can further enhance security by preventing attackers from making this independent validation when only equipped with less powerful computers. [ABSTRACT FROM AUTHOR]

In this work, we provide a variation of the four-circulant construction for self-dual codes. By applying the constructions over the alphabets F2, F2 + uF2, F4 + uF4, 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 new parameters have been constructed. [ABSTRACT FROM AUTHOR]

We describe a bordered construction for self-dual codes coming from group rings. We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary self-dual codes of various lengths. In particular we find a new extremal binary self-dual code of length 78. [ABSTRACT FROM AUTHOR]

We study G-codes over the ring Z4, which are codes that are held invariant by the action of an arbitrary group G. We view these codes as ideals in a group ring and we study the rank and kernel of these codes. We use the rank and kernel to study the image of these codes under the Gray map. We study the specific case when the group is the dihedral group and the dicyclic group. Finally, we study quaternary self-dual dihedral and dicyclic codes, tabulating the many good self-dual quaternary codes obtained via these constructions, including the octacode. [ABSTRACT FROM AUTHOR]

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 F2 + uF2 and F4 + uF4, we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally self-dual codes with parameters [72,36,14]. [ABSTRACT FROM AUTHOR]

In this work, construction methods for formally self-dual codes are generalized in the form of block λ-circulant matrices. The constructions are applied over the rings F2, R1 = F2 + uF2 and S = F2[u]/(u³ - 1). Using n-block λ-circulant matrices for suitable integers n and units λ, 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 twenty-five new near-extremal FSD even codes of length 36. [ABSTRACT FROM AUTHOR]

In this work, we find a form for the homogeneous weight over the ring Rk,m, using the related theoretical results from the literature. We then use the first order Reed-Muller codes to find a distance-preserving map that takes codes over Rk,m to binary codes. By considering cyclic, constacyclic and quasicyclic codes over Rk;m of different lengths for different values of k and m, we construct a considerable number of optimal binary codes that are divisible with high levels of divisibility. The codes we have obtained are also quasicyclic with high indices and they are all self-orthogonal when km ≥ 4. The results, which have been obtained by computer search are tabulated. [ABSTRACT FROM AUTHOR]

In this work, we focus on reversible cyclic codes which correspond to reversible DNA codes or reversible-complement DNA codes over a family of finite chain rings, in an effort to extend what was done by Yildiz and Siap in [20]. The ring family that we have considered are of size 22k , k = 1, 2, . . . and we match each ring element with a DNA 2k-1-mer. We use the so-called u2-adic digit system to solve the reversibility problem and we characterize cyclic codes that correspond to reversible-complement DNA-codes. We then conclude our study with some examples. [ABSTRACT FROM AUTHOR]

In this paper, we use the graded ring construction to lift the extended binary Hamming code of length 8 to $$R_k$$ . Using this method we construct self-dual codes over $$R_3$$ of length 8 whose Gray images are self-dual binary codes of length 64. In this way, we obtain twenty six non-equivalent extremal binary Type I self-dual codes of length 64, ten of which have weight enumerators that were not previously known to exist. The new codes that we found have $$\beta = 1, 5, 13, 17, 21, 25, 29, 33, 41$$ and 52 in $$W_{64,2}$$ and they all have automorphism groups of size 8. [ABSTRACT FROM AUTHOR]

In this paper skew cyclic codes over the the family of rings Fq+vFq with v2 = v are studied for the first 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 first time in literature. Moreover, a BCH type bound is presented for the parameters of these codes. [ABSTRACT FROM AUTHOR]

The focus of this work is constructing Gray maps for linear codes over a family of Frobenius rings, using tools from combinatorial geometries. The main combinatorial structure that are used are projective geometries PGn(q), defined over Fq. Codes over Zpk , Galois rings, finite chain rings and Rk have been considered with respect to homogeneous weights. Using hyperplanes in projective geometries, distance preserving Gray maps from aforementioned rings to residue fields have been constructed. The use of projective geometries serves as a novel approach to what was a primarily algebraic approach in the literature. The constructions also serve as a motivation for using homogeneous weights. [ABSTRACT FROM AUTHOR]

We consider quasi-cyclic codes over the ring $$\mathbb{F }_2+u\mathbb{F }_2+v\mathbb{F }_2+uv\mathbb{F }_2$$ , a finite non-chain ring that has been recently studied in coding theory. The Gray images of these codes are shown to be binary quasi-cyclic codes. Using this method we have obtained seventeen new binary quasi-cyclic codes that are new additions to the database of binary quasi-cyclic codes. Moreover, we also obtain a number of binary quasi-cyclic codes with the same parameters as best known binary linear codes that otherwise have more complicated constructions. [ABSTRACT FROM AUTHOR]

A lift of binary self-dual codes to the ring R2 is described. By using this lift, a family of self-dual codes over R2of length 17 are constructed. Taking the binary images of these codes, extremal binary self-dual codes of length 68 are obtained. For the first time in the literature, extremal binary codes of length 68 with γ= 4 andγ= 6 in W68;2 have been obtained. In addition to these, six new codes withγ= 0 and fourteen new codes with γ= 2 in W68;2 have also been found. [ABSTRACT FROM AUTHOR]

We study self-dual codes over an infinite family of rings, denoted Rk, which has been recently introduced to the literature. We prove that for each self-dual code over Rk, k ≥ 2, there exist a corresponding binary self-dual code, a real unimodular lattice, a complex unimodular lattice, a quaternionic lattice and an infinite family of self-dual codes. We prove the existence of Type II codes of all lengths over Rk, for k ≥ 3, and we obtain some extremal binary self-dual codes including the extended binary Golay code as the Gray images of self-dual codes over Rk for some suitable k. The binary self-dual codes obtained from Rk all have automorphism groups whose orders are a multiple of 2k. [ABSTRACT FROM AUTHOR]