Gröbner basis for norm-trace codes

The root diagram for one-point AG codes arising from certain curves with separated variables

“…we have the following algorithm, which is completely analogous to Proposition 4.4 in [12] and Algorithm 4.2 in [4].…”

“…But, in [11], using an appropriate automorphism of the Hermitian curve, Little et al introduced the concept of root diagram that allows to construct an algorithm for computing a Gröbner basis with a lower complexity for one-point Hermitian codes. In [4], the results of [11] were extended to codes arising from the norm-trace curve, which is a generalization of the Hermitian curve. In both works, the one-point AG codes arising from curves over finite fields F q with q elements and the construction of the root diagram is made by using automorphisms whose order is equal to q − 1.…”

“…In other words, the root diagram is the key to the construction of the algorithm given in [12,Proposition 4.4]. In [4], the results of [12] were extended to codes arising from the norm-trace curve, which is a generalization of the Hermitian curve. In this work, using the same techniques used in [12] and [4], we will construct the root diagram for codes arising from certain curves, which we will denote by X , with separated variables that has certain special automorphism and a Weierstrass semigroup generated by two elements (see (1), (2) and (3) in Section 3).…”

“…In [4], the results of [12] were extended to codes arising from the norm-trace curve, which is a generalization of the Hermitian curve. In this work, using the same techniques used in [12] and [4], we will construct the root diagram for codes arising from certain curves, which we will denote by X , with separated variables that has certain special automorphism and a Weierstrass semigroup generated by two elements (see (1), (2) and (3) in Section 3). In addition to Hermitian and norm-trace curves, we have important examples of such curves X with recent applications in coding theory, namely: the maximal curve with plane model y q + y = x q r +1 , see [11]; a quotient of the Hermitian curve with plane model y q + y = x m , see [13]; and curves on Kummer extensions, see [2].…”

On generator and parity-check polynomial matrices of generalized quasi-cyclic codes

“…A GQC code is equivalent to a linear code with a non-trivial automorphism group, where its cyclic subgroup is fixed [9]. Many of algebraic geometry codes, especially Hermitian codes, are GQC codes because these codes have non-trivial automorphisms [6,20,21]. A subclass of finite geometry codes, especially projective geometry codes, is not QC codes but GQC codes, and there are many high performance LDPC codes in GQC codes [13,29].…”