Bivariate Linear Operator Codes
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908382234935296 |
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| author | Putterman, Aaron L. Zaripov, Vadim |
| author_facet | Putterman, Aaron L. Zaripov, Vadim |
| contents | In this work, we present a generalization of the linear operator family of codes that captures more codes that achieve list decoding capacity. Linear operator (LO) codes were introduced by Bhandari, Harsha, Kumar, and Sudan [BHKS24] as a way to capture capacity-achieving codes. In their framework, a code is specified by a collection of linear operators that are applied to a message polynomial and then evaluated at a specified set of evaluation points. We generalize this idea in a way that can be applied to bivariate message polynomials, getting what we call bivariate linear operator (B-LO) codes.
We show that bivariate linear operator codes capture more capacity-achieving codes, including permuted product codes introduced by Berman, Shany, and Tamo [BST24]. These codes work with bivariate message polynomials, which is why our generalization is necessary to capture them as a part of the linear operator framework.
Similarly to the initial paper on linear operator codes, we present sufficient conditions for a bivariate linear operator code to be list decodable. Using this characterization, we are able to derive the theorem characterizing list-decodability of LO codes as a specific case of our theorem for B-LO codes. We also apply this theorem to show that permuted product codes are list decodable up to capacity, thereby unifying this result with those of known list-decodable LO codes, including Folded Reed-Solomon, Multiplicity, and Affine Folded Reed-Solomon codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16596 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bivariate Linear Operator Codes Putterman, Aaron L. Zaripov, Vadim Information Theory In this work, we present a generalization of the linear operator family of codes that captures more codes that achieve list decoding capacity. Linear operator (LO) codes were introduced by Bhandari, Harsha, Kumar, and Sudan [BHKS24] as a way to capture capacity-achieving codes. In their framework, a code is specified by a collection of linear operators that are applied to a message polynomial and then evaluated at a specified set of evaluation points. We generalize this idea in a way that can be applied to bivariate message polynomials, getting what we call bivariate linear operator (B-LO) codes. We show that bivariate linear operator codes capture more capacity-achieving codes, including permuted product codes introduced by Berman, Shany, and Tamo [BST24]. These codes work with bivariate message polynomials, which is why our generalization is necessary to capture them as a part of the linear operator framework. Similarly to the initial paper on linear operator codes, we present sufficient conditions for a bivariate linear operator code to be list decodable. Using this characterization, we are able to derive the theorem characterizing list-decodability of LO codes as a specific case of our theorem for B-LO codes. We also apply this theorem to show that permuted product codes are list decodable up to capacity, thereby unifying this result with those of known list-decodable LO codes, including Folded Reed-Solomon, Multiplicity, and Affine Folded Reed-Solomon codes. |
| title | Bivariate Linear Operator Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2411.16596 |