New Galois hulls of generalized Reed-Solomon codes

Most recently, Gao et al. found a nice method to investigate the Euclidean hulls of generalized Reed-Solomon codes in terms of Goppa codes. In this note, we extend the results to general Galois hull. We prove that the Galois hulls of some GRS codes are still GRS codes. We also give some examples on Galois LCD and self-dual MDS codes. Compare with known results, the Galois hulls of GRS codes obtained in this work have flexible parameters.     

A Subclass of Binary Goppa Codes With Improved Estimation of the Code Dimension

A new bound for the dimension of binary Goppa codes belonging to a specific subclass is given. This bound improves the well-known lower bound for Goppa codes.     

Negacyclic codes over Galois rings of characteristic 2 a

We investigate negacyclic codes over the Galois ring GR(2 a ,m) of length N = 2 k n,where n is odd and k 0.We first determine the structure of u-constacyclic codes of length n over the finite chain ring GR(2 a ,m)[u]/ u 2 k + 1 .Then using a ring isomorphism we obtain the structure of negacyclic codes over GR(2 a ,m) of length N = 2 k n (n odd) and explore the existence of self-dual negacyclic codes over GR(2 a ,m).A bound for the homogeneous distance of such negacyclic codes is also given.     

Goppa codes over the p-adic integers and integers modulo pe

Goppa codes were defined by Valery D. Goppa in 1970. In 1978, Robert J. McEliece used this family of error-correcting codes in his cryptosystem, which has gained popularity in the last decade due to its resistance to attacks from quantum computers. In this paper, we present Goppa codes over the p-adic integers and integers modulo $p^e$. This allows the creation of chains of Goppa codes over different rings. We show some of their properties, such as parity-check matrices and minimum distance, and suggest their cryptographic application, following McEliece's scheme.     

Algebraic geometric codes with applications

The theory of linear error-correcting codes from algebraic geometric curves (algebraic geometric (AG) codes or geometric Goppa codes) has been well-developed since the work of Goppa and Tsfasman, Vladut, and Zink in 1981–1982. In this paper we introduce to readers some recent progress in algebraic geometric codes and their applications in quantum error-correcting codes, secure multi-party computation and the construction of good binary codes.      

On Hermitian LCD codes and their Gray image

We provide methods and algorithms to construct Hermitian linear complementary dual (LCD) codes over finite fields. We study existence of self-dual basis with respect to Hermitian inner product, and as an application, we construct Euclidean LCD codes by projecting the Hermitian codes over such a basis. Many optimal quaternary Hermitian and ternary Euclidean LCD codes are obtained. Comparisons with classical constructions are made.     

The New Minimum Distance Bounds of Goppa Codes and Their Decoding

A couple of new lower bounds of the minimum distance of Goppa codes is derived, using an extended field code for a Goppa code which contains the Goppa code as its subfield-subcode. Also presented are procedures for both error-only and error-and-erasure decoding for Goppa codes up to the new lower bounds, based on the Berlekamp-Massey algorithm and the Feng-Tzeng multisequence shift-register synthesis algorithms which have been used for decoding cyclic codes up to the BCH and HT(Hartmann-Tzeng) bounds.     

On Z2Z4-additive complementary dual codes and related LCD codes

Linear complementary dual codes were defined by Massey in 1992, and were used to give an optimum linear coding solution for the two user binary adder channel. In this paper, we define the analog of LCD codes over fields in the ambient space with mixed binary and quaternary alphabets. These codes are additive, in the sense that they are additive subgroups, rather than linear as they are not vector spaces over some finite field. We study the structure of these codes and we use the canonical Gray map from this space to the Hamming space to construct binary LCD codes in certain cases. We give examples of such binary LCD codes which are distance-optimal, i.e., they have the largest minimum distance among all binary LCD codes with the same length and dimension.     

Self-reciprocal and self-conjugate-reciprocal irreducible factors of xn − λ and their applications

In this paper, we present some necessary and sufficient conditions under which an irreducible polynomial is self-reciprocal (SR) or self-conjugate-reciprocal (SCR). By these characterizations, we obtain some enumeration formulas of SR and SCR irreducible factors of $x^n-\lambda$, $\lambda \in {\mathbb{F}}_q^{⁎}$, over ${\mathbb{F}}_q$, which are just open questions posed by Boripan et al. (2019). We also count the numbers of Euclidean and Hermitian LCD constacyclic codes and show some well-known results on Euclidean and Hermitian self-dual constacyclic codes in a simple and direct way.     

Quantum MDS codes with new length and large minimum distance

According to the well-known CSS construction, constructing quantum MDS codes are extensively investigated via Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. In this paper, given two Hermitian self-orthogonal GRS codes $GR{S}_{k_1}(A,{\mathit{v}}_A)$ and $GR{S}_{k_2}(B,{\mathit{v}}_B)$, we propose a sufficient condition to ensure that $GR{S}_k(A\cup B,{\mathit{v}}_{A\cup B})$ is still a Hermitian self-orthogonal code. Consequently, we first present a new general construction of infinitely families of quantum MDS codes from known ones. Moreover, applying the trace function and norm function over finite fields, we give another two new constructions of quantum MDS codes with flexible parameters. It turns out that the forms of the lengths of our quantum MDS codes are quite different from previous known results in the literature. Meanwhile, the minimum distances of all the q-ary quantum MDS codes are bigger than $q/2+1$.     

Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings

Let $R$ be the Galois ring $GR(p^k,s)$ of characteristic $p^k$ and cardinality $p^{sk}$. Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length $n$ over $R$, and a $p$-adic integer ring with $gcd(p,n)=1$. Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length $r{l}^m$ over $R$, where $r,l,$ and $t$ are three primes with $2?l$, $r|(q^t?1)$, $l^v\parallel (q^t?1)$ and $gcd(rl,q(q?1))=1$. Finally, as applications, weight distributions of all irreducible cyclic codes for $t=2$ and generator polynomials of self-dual cyclic codes of length $l^m$ and $r{l}^m$ over $R$ are given.     

Binary construction of quantum codes of minimum distances five and six

In this paper, we construct a large number of good quantum codes of minimum distances five and six by Steane's Construction. Our methods involve the study of the check matrices of binary extended BCH-codes, together with puncturing and combining such matrices.