Upper Bounds on the Minimum Distance of Structured LDPC Codes
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| Format: | Preprint |
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2025
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| _version_ | 1866912213015461888 |
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| author | Arnault, François Gaborit, Philippe Rozendaal, Wouter Saussay, Nicolas Zémor, Gilles |
| author_facet | Arnault, François Gaborit, Philippe Rozendaal, Wouter Saussay, Nicolas Zémor, Gilles |
| contents | We investigate the minimum distance of structured binary Low-Density Parity-Check (LDPC) codes whose parity-check matrices are of the form $[\mathbf{C} \vert \mathbf{M}]$ where $\mathbf{C}$ is circulant and of column weight $2$, and $\mathbf{M}$ has fixed column weight $r \geq 3$ and row weight at least $1$. These codes are of interest because they are LDPC codes which come with a natural linear-time encoding algorithm. We show that the minimum distance of these codes is in $O(n^{\frac{r-2}{r-1} + ε})$, where $n$ is the code length and $ε> 0$ is arbitrarily small. This improves the previously known upper bound in $O(n^{\frac{r-1}{r}})$ on the minimum distance of such codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_19125 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper Bounds on the Minimum Distance of Structured LDPC Codes Arnault, François Gaborit, Philippe Rozendaal, Wouter Saussay, Nicolas Zémor, Gilles Information Theory We investigate the minimum distance of structured binary Low-Density Parity-Check (LDPC) codes whose parity-check matrices are of the form $[\mathbf{C} \vert \mathbf{M}]$ where $\mathbf{C}$ is circulant and of column weight $2$, and $\mathbf{M}$ has fixed column weight $r \geq 3$ and row weight at least $1$. These codes are of interest because they are LDPC codes which come with a natural linear-time encoding algorithm. We show that the minimum distance of these codes is in $O(n^{\frac{r-2}{r-1} + ε})$, where $n$ is the code length and $ε> 0$ is arbitrarily small. This improves the previously known upper bound in $O(n^{\frac{r-1}{r}})$ on the minimum distance of such codes. |
| title | Upper Bounds on the Minimum Distance of Structured LDPC Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2501.19125 |