Algebraic Expander Codes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908914012913664 |
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| author | Kopparty, Swastik Tamo, Itzhak |
| author_facet | Kopparty, Swastik Tamo, Itzhak |
| contents | Expander (Tanner) codes combine sparse graphs with local constraints, enabling linear-time decoding and asymptotically good distance--rate tradeoffs. A standard constraint-counting argument yields the global-rate lower bound $R\ge 2r-1$ for a Tanner code with local rate $r$, which gives no positive-rate guarantee in the low-rate regime $r\le 1/2$. This regime is nonetheless important in applications that require algebraic local constraints (e.g., Reed--Solomon locality and the Schur-product/multiplication property).
We introduce \emph{Algebraic Expander Codes}, an explicit algebraic family of Tanner-type codes whose local constraints are Reed--Solomon and whose global rate remains bounded away from $0$ for every fixed $r\in(0,1)$ (in particular, for $r\le 1/2$), while achieving constant relative distance.
Our codes are defined by evaluating a structured subspace of polynomials on an orbit of a non-commutative subgroup of $\mathrm{AGL}(1,\mathbb{F})$ generated by translations and scalings. The resulting sparse coset geometry forms a strong spectral expander, proved via additive character-sum estimates, while the rate analysis uses a new notion of polynomial degree and a polytope-volume/dimension-counting argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24788 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic Expander Codes Kopparty, Swastik Tamo, Itzhak Information Theory Expander (Tanner) codes combine sparse graphs with local constraints, enabling linear-time decoding and asymptotically good distance--rate tradeoffs. A standard constraint-counting argument yields the global-rate lower bound $R\ge 2r-1$ for a Tanner code with local rate $r$, which gives no positive-rate guarantee in the low-rate regime $r\le 1/2$. This regime is nonetheless important in applications that require algebraic local constraints (e.g., Reed--Solomon locality and the Schur-product/multiplication property). We introduce \emph{Algebraic Expander Codes}, an explicit algebraic family of Tanner-type codes whose local constraints are Reed--Solomon and whose global rate remains bounded away from $0$ for every fixed $r\in(0,1)$ (in particular, for $r\le 1/2$), while achieving constant relative distance. Our codes are defined by evaluating a structured subspace of polynomials on an orbit of a non-commutative subgroup of $\mathrm{AGL}(1,\mathbb{F})$ generated by translations and scalings. The resulting sparse coset geometry forms a strong spectral expander, proved via additive character-sum estimates, while the rate analysis uses a new notion of polynomial degree and a polytope-volume/dimension-counting argument. |
| title | Algebraic Expander Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2603.24788 |