Quasi-Perfect and Distance-Optimal Codes Sum-Rank Codes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917590705635328 |
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| author | Chen, Hao |
| author_facet | Chen, Hao |
| contents | Constructions of distance-optimal codes and quasi-perfect codes are challenging problems and have attracted many attentions. In this paper, we give the following three results.
1) If $λ|q^{sm}-1$ and $λ<\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+ε)}}$, an infinite family of distance-optimal $q$-ary cyclic sum-rank codes with the block length $t=\frac{q^{sm}-1}λ$, the matrix size $s \times s$, the cardinality $q^{s^2t-s(2m+3)}$ and the minimum sum-rank distance four is constructed.
2) Block length $q^4-1$ and the matrix size $2 \times 2$ distance-optimal sum-rank codes with the minimum sum-rank distance four and the Singleton defect four are constructed. These sum-rank codes are close to the sphere packing bound , the Singleton-like bound and have much larger block length $q^4-1>>q-1$.
3) For given positive integers $m$ satisfying $2 \leq m$, an infinite family of quasi-perfect sum-rank codes with the matrix size $2 \times m$, and the minimum sum-rank distance three is also constructed. Quasi-perfect binary sum-rank codes with the minimum sum-rank distance four are also given.
Almost MSRD $q$-ary codes with the block lengths up to $q^2$ are given. We show that more distance-optimal binary sum-rank codes can be obtained from the Plotkin sum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11160 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasi-Perfect and Distance-Optimal Codes Sum-Rank Codes Chen, Hao Information Theory Constructions of distance-optimal codes and quasi-perfect codes are challenging problems and have attracted many attentions. In this paper, we give the following three results. 1) If $λ|q^{sm}-1$ and $λ<\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+ε)}}$, an infinite family of distance-optimal $q$-ary cyclic sum-rank codes with the block length $t=\frac{q^{sm}-1}λ$, the matrix size $s \times s$, the cardinality $q^{s^2t-s(2m+3)}$ and the minimum sum-rank distance four is constructed. 2) Block length $q^4-1$ and the matrix size $2 \times 2$ distance-optimal sum-rank codes with the minimum sum-rank distance four and the Singleton defect four are constructed. These sum-rank codes are close to the sphere packing bound , the Singleton-like bound and have much larger block length $q^4-1>>q-1$. 3) For given positive integers $m$ satisfying $2 \leq m$, an infinite family of quasi-perfect sum-rank codes with the matrix size $2 \times m$, and the minimum sum-rank distance three is also constructed. Quasi-perfect binary sum-rank codes with the minimum sum-rank distance four are also given. Almost MSRD $q$-ary codes with the block lengths up to $q^2$ are given. We show that more distance-optimal binary sum-rank codes can be obtained from the Plotkin sum. |
| title | Quasi-Perfect and Distance-Optimal Codes Sum-Rank Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2401.11160 |