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# Some classes of LCD codes and self-orthogonal codes over finite fields

• * Corresponding author: Chunming Tang
• Due to their important applications in theory and practice, linear complementary dual (LCD) codes and self-orthogonal codes have received much attention in the last decade. The objective of this paper is to extend a recent construction of binary LCD codes and self-orthogonal codes to the general $p$-ary case, where $p$ is an odd prime. Based on the extended construction, several classes of $p$-ary linear codes are obtained. The characterizations of these linear codes to be LCD or self-orthogonal are derived. The duals of these linear codes are also studied. It turns out that the proposed linear codes are optimal in many cases in the sense that their parameters meet certain bounds on linear codes. The weight distributions of these linear codes are settled.

Mathematics Subject Classification: Primary: 06E75, 94A60, 11T23.

 Citation: • • Table 1.  The weight distribution of ${\mathcal C}_{D_{t}}$ for $1\leq t\leq m$

 Weight Multiplicity $0$ $1$ time $\frac{(p-1)n-(p-1)K_{t}(k,m)}{p}$ for $k=1,2,\cdots,m-1$ $(p-1)^k{m\choose k}$ times $\frac{(p-1){m\choose t}((p-1)^t+(-1)^{t+1})}{p}$ $(p-1)^m$ times

Table 2.  The weight distribution of ${\mathcal C}_{D_{t}}$ for $1\leq t\leq m$

 Weight Multiplicity $0$ $1$ time $\frac{\sum^t\limits_{i=1}(p-1)^{i+1}{{m}\choose{i}}-(p-1)^{t+1}{{m-1}\choose{t}}+p-1}{p}$ $(p-1)m$ times $\frac{(p-1)n-(p-1)K_{t}(k-1,m-1)+(p-1)}{p}$ $k=2,\cdots,m$ $(p-1)^k{m\choose k}$ times

Table 3.  The weight distribution of $\mathcal C_{\overline{D}_ t}$ for $1\leq t\leq m$

 Weight Multiplicity $0$ $1$ time $\frac{(p-1)n-(p-1) \left ( K_{t}(k,m)+ K_{m}(k,m) \right )}{p}$ for $k=1,2,\cdots,m-1$ $(p-1)^k{m\choose k}$ times $\frac{(p-1){m\choose t}((p-1)^t+(-1)^{t+1})+(p-1)((p-1)^m-(-1)^m)}{p}$ $(p-1)^m$ times

Table 4.  The weight distribution of ${\mathcal C}_{\overline{D}_{t}}$ for $1\leq t\leq m$

 Weight Multiplicity $0$ $1$ time $\frac{p(p-1)^{m}+\sum^t\limits_{i=1}(p-1)^{i+1}{{m}\choose{i}}-(p-1)^{t+1}{{m-1}\choose{t}}+p-1}{p}$ $(p-1)m$ times $\frac{(p-1)n-(p-1)\left ( K_{t}(k-1,m-1)+K_{m}(k,m) -1\right )}{p}$ $k=2,\cdots,m$ $(p-1)^k{m\choose k}$ times
• Tables(4)

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