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### Revision(s):

Revision #1 to TR22-069 | 6th May 2022 07:43

#### List-Decoding XOR Codes Near the Johnson Bound

Revision #1
Authors: Silas Richelson, Sourya Roy
Accepted on: 6th May 2022 07:43
Keywords:

Abstract:

In a breakthrough result, Ta-Shma described an explicit construction of an almost optimal binary code (STOC 2017). Ta-Shma's code has distance $\frac{1-\varepsilon}{2}$ and rate $\Omega\bigl(\varepsilon^{2+o(1)}\bigr)$ and thus it almost achieves the Gilbert-Varshamov bound, except for the $o(1)$ term in the exponent. The prior best list-decoding algorithm for (a variant of) Ta-Shma's code achieves is due to Alev et al (STOC 2021). This algorithm makes use of SDP hierarchies, and is able to recover from a $\frac{1-\rho}{2}-$fraction of errors as long as $\rho\geq2^{\log(1/\varepsilon)^{1/6}}$. In this work we give an improved analysis of a similar list-decoding algorithm. Our algorithm works for Ta-Shma's original code, and it is able to list-decode almost all the way to the Johnson bound: it can recover from a $\frac{1-\rho}{2}-$fraction of errors as long as $\rho\geq2\sqrt{\varepsilon}$.

### Paper:

TR22-069 | 28th April 2022 11:30

#### List-Decoding Random Walk XOR Codes Near the Johnson Bound

TR22-069
Authors: Silas Richelson, Sourya Roy
Publication: 5th May 2022 16:45
Keywords:

Abstract:

In a breakthrough result, Ta-Shma described an explicit construction of an almost optimal binary code (STOC 2017). Ta-Shma's code has distance $\frac{1-\varepsilon}{2}$ and rate $\Omega\bigl(\varepsilon^{2+o(1)}\bigr)$ and thus it almost achieves the Gilbert-Varshamov bound, except for the $o(1)$ term in the exponent. The prior best list-decoding algorithm for (a variant of) Ta-Shma's code achieves is due to Alev et al (STOC 2021). This algorithm makes use of SDP hierarchies, and is able to recover from a $\frac{1-\rho}{2}-$fraction of errors as long as $\rho\geq2^{\log(1/\varepsilon)^{1/6}}$. In this work we give an improved analysis of a similar list-decoding algorithm. Our algorithm works for Ta-Shma's original code, and it is able to list-decode almost all the way to the Johnson bound: it can recover from a $\frac{1-\rho}{2}-$fraction of errors as long as $\rho\geq2\sqrt{\varepsilon}$.

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