Goppa Codes Over Certain Semigroups

IEEE Transactions on Information Theory, 1975

2021

Generalized Goppa codes are defined by a code locator set ℒ of polynomials and a Goppa polynomial G(x). When the degree of all code locator polynomials in ℒ is one, generalized Goppa codes are classical Goppa codes. In this work, binary generalized Goppa codes are investigated. First, a parity-check matrix for these codes with code locators of any degree is derived. A careful selection of the code locators leads to a lower bound on the minimum Hamming distance of generalized Goppa codes which improves upon previously known bounds. A quadratic-time decoding algorithm is presented which can decode errors up to half of the minimum distance. Interleaved generalized Goppa codes are introduced and a joint decoding algorithm is presented which can decode errors beyond half the minimum distance with high probability. Finally, some code parameters and how they apply to the Classic McEliece post-quantum cryptosystem are shown.

Designs, Codes and Cryptography, 2018

A class of q-ary totally decomposed cumulative Γ (L, G j)-codes with L = {α ∈: G(α) = 0} and G j = G j (x), 1 ≤ j ≤ q, where G(x) is a polynomial totally decomposed in G F(q m), are considered. The relation between different codes from this class is studied. Improved bounds of the minimum distance and dimension are obtained. Keywords Goppa codes • Totally decomposed Goppa codes • Wild Goppa codes Mathematics Subject Classification 94B05 • 94B65 This is one of several papers published in Designs, Codes and Cryptography comprising the "Special Issue on Coding and Cryptography".

TEMA - Tendências em Matemática Aplicada e Computacional, 2011

In this paper, we introduced new construction techniques of BCH, alternant, Goppa, Srivastava codes through the semigroup ring B[X; 1 3 Z0] instead of the polynomial ring B[X; Z0], where B is a finite commutative ring with identity, and for these constructions we improve the several results of [1]. After this, we present a decoding principle for BCH, alternant and Goppa codes which is based on modified Berlekamp-Massey algorithm. This algorithm corrects all errors up to the Hamming weight t ≤ r/2, i.e., whose minimum Hamming distance is r + 1.

Computational & Applied Mathematics, 2011

This paper introduces novel constructions of cyclic codes using semigroup rings instead of polynomial rings. These constructions are applied to define and investigate the BCH, alternant, Goppa, and Srivastava codes. This makes it possible to improve several recent results due to Andrade and Palazzo [1].

Discrete Applied Mathematics, 2006

We consider irreducible Goppa codes over Fq of length q n defined by polynomials of degree r where q is a prime power and n, r are arbitrary positive integers. We obtain an upper bound on the number of such codes.

2019

We obtain upper bounds on the number of irreducible and extended irreducible Goppa codes over $GF(p)$ of length $q$ and $q+1$, respectively defined by polynomials of degree $r$, where $q=p^t$ and $r\geq 3$ is a positive integer.

International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings., 2004

Let q be a power of a prime p, n an arbitrary integer and g an irreducible polynomial over Fqn = L of degree r. Let (u, v) denote gcd (u, v). We show that the condition p|r is a sufficient condition for the the Goppa code Γ(L, g) over Fq to be quasicyclic and we give a method for the construction of these quasicyclic codes. We also construct quasicyclic Goppa codes of degree r over Fq of length q n − 1 where n, r are integers such that a non trivial factor s of nr divides q n − 1, every prime dividing s is a divisor of r and n nr s , n , r = 1. Again these conditions are sufficient for the existence of these quasicyclic Goppa codes.

Anais do XXI Simpósio Brasileiro de Telecomunicações

Goppa and Srivastava have described interesting classes of linear noncyclic error-correcting codes over finite fields. In this work we present a construction technique of Goppa and Srivastava codes over local finite commutative rings with identity in terms of parity-check matrix and an efficient decoding procedure, based on the modified Berlekamp-Massey algorithm, is proposed for the Goppa codes.

IEEE Transactions on Information Theory, 1992

A research problem in The Theory of Error Correcting Codes by MacWilliams and Sloane asks for the true dimension and the minimum distance of a Goppa Code. The problem is solved for the minimum distance whenever the length of the code satisfies a certain inequality on the degree of the Goppa polynomial. In order to do this, conditions are improved on a theorem of E. Bombieri. This improvement is used also to generalize a previous result on the minimum distance of the dual of a Goppa code. This approach is generalized and results are obtained about the parameters of a class of subfield subcodes of geometric Goppa codes; in order words, the covering radius are estimated, and further, the number of information symbols whenever the minimum distance is small in relation to the length of the code are found. Finally, a bound on the minimum distance of the dual code is discussed.