A New Class of Linear Codes
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917777395154944 |
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| author | Cherubini, Giacomo Micheli, Giacomo |
| author_facet | Cherubini, Giacomo Micheli, Giacomo |
| contents | Let $n$ be a prime power, $r$ be a prime with $r\mid n-1$, and $\varepsilon\in (0,1/2)$. Using the theory of multiplicative character sums and superelliptic curves, we construct new codes over $\mathbb F_r$ having length $n$, relative distance $(r-1)/r+O(n^{-\varepsilon})$ and rate $n^{-1/2-\varepsilon}$. When $r=2$, our binary codes have exponential size when compared to all previously known families of linear and non-linear codes with relative distance asymptotic to $1/2$, such as Delsarte--Goethals codes. Moreover, concatenating with a Reed--Solomon code gives a family of codes of length $n$, asymptotic distance $1/2$ and rate $Ω(n^{-\varepsilon})$ for any fixed small $\varepsilon>0$, improving our initial construction. Such rate is also asymptotically better than the one by Kschischang and Tasbihi obtained by concatenating a Reed--Solomon with Reed--Muller, improving by a factor in $Ω(n^{1/2}/\log(n))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A New Class of Linear Codes Cherubini, Giacomo Micheli, Giacomo Information Theory Number Theory 11T06, 11T71, 68P30 Let $n$ be a prime power, $r$ be a prime with $r\mid n-1$, and $\varepsilon\in (0,1/2)$. Using the theory of multiplicative character sums and superelliptic curves, we construct new codes over $\mathbb F_r$ having length $n$, relative distance $(r-1)/r+O(n^{-\varepsilon})$ and rate $n^{-1/2-\varepsilon}$. When $r=2$, our binary codes have exponential size when compared to all previously known families of linear and non-linear codes with relative distance asymptotic to $1/2$, such as Delsarte--Goethals codes. Moreover, concatenating with a Reed--Solomon code gives a family of codes of length $n$, asymptotic distance $1/2$ and rate $Ω(n^{-\varepsilon})$ for any fixed small $\varepsilon>0$, improving our initial construction. Such rate is also asymptotically better than the one by Kschischang and Tasbihi obtained by concatenating a Reed--Solomon with Reed--Muller, improving by a factor in $Ω(n^{1/2}/\log(n))$. |
| title | A New Class of Linear Codes |
| topic | Information Theory Number Theory 11T06, 11T71, 68P30 |
| url | https://arxiv.org/abs/2401.07986 |