Huimin Lao

Yaqi Chen, Hao Chen, Cunsheng Ding et al.

Goppa codes form an important class of alternant codes with wide applications in algebraic coding theory and code-based cryptography. Determining the true minimum distance of a Goppa code is a difficult problem. In this paper, we provide a necessary and sufficient criterion for a Goppa code to attain its designed distance $δ=t+1$, where $t$ is the degree of the Goppa polynomial. As applications, we determine the minimum distances of several classes of $q$-ary Goppa codes. In particular, we prove the tightness of the improved lower bound for a class of wild Goppa codes, and extend the family with $G(x)=x^t+A$ from the binary case to arbitrary odd prime powers. We then specialize the criterion to the monomial case $G(x)=x^t$, which is equivalent to primitive BCH codes. This leads to several infinite families of primitive BCH codes with $d=δ$, including the binary codes $\mathbf{C}_{(2,2^m-1,9,1)}$ and $\mathbf{C}_{(2,2^m-1,15,1)}$, the family $\mathbf{C}_{(p,p^p-1,2p+2,1)}$ with an odd prime $p$ and the family $\mathbf{C}_{(q,q^m-1,r\frac{q^m-1}{q-1}+1,1)}$ with $r\mid q-1$. In particular, we prove that the primitive BCH code $\mathbf{C}_{(q,q^m-1,q^t+1,1)}$ has minimum distance $q^t+1$ under the condition $t\mid m$, improving the previously known condition $pt\mid m$.

Yaqi Chen, Hao Chen, Cunsheng Ding et al.

BCH codes form an important class of cyclic codes, which have applications in communication and data storage systems. Although the BCH bound provides a lower bound on the minimum distance of BCH codes, determining the true minimum distances of BCH codes is a very challenging problem. In this paper, we settle the minimum distances of a number of infinite families of narrow-sense BCH codes. By explicitly constructing the locator polynomials for minimum weight codewords, we obtain many families of primitive and non-primitive BCH codes with $d=δ$, where $d$ is the minimum distance of a $q$-ary BCH code of length $n$, designed distance $δ$, and offset $b$, denoted by $\mathbf{C}_{(q, n, δ, b)}$. For primitive BCH codes, we obtain infinite families of BCH codes over $\mathbb{F}_3$ and $\mathbb{F}_4$ satisfying $d=δ$, where $δ\in \{5,6,7,8\}$. Moreover, we construct several infinite families of $q$-ary BCH codes with $d=δ$, where $2 \le δ\le q-1$. For $δ=q^t+1$, we prove that the BCH code $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ has $d=δ$ for all $m$ satisfying $m \equiv 0 \pmod{pt}$, where $p$ denotes the characteristic of $\mathbb{F}_q$. In the paper by Ding et al., IEEE Trans. Inf. Theory 61(5): 2351-2356, it was conjectured that the minimum distance of $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ is always equal to its Bose distance $d_B$. Our result confirms this conjecture for the case $m \equiv 0 \pmod{pt}$. For non-primitive BCH codes, we construct a family of BCH codes $\mathbf{C}_{(q,\frac{q^p-1}λ,p+1,1)}$ with $d=δ=p+1$, where $p$ is an odd prime, $q=p^e$ with $p \nmid e$ and $λ\mid q-1$.