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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2411.14361 |
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| _version_ | 1866913582774484992 |
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| author | Basu, Arpon Hsieh, Jun-Ting Kothari, Pravesh K. Lin, Andrew D. |
| author_facet | Basu, Arpon Hsieh, Jun-Ting Kothari, Pravesh K. Lin, Andrew D. |
| contents | We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm1\}^k \rightarrow \{\pm1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in a sequence of prior works. For $q=3$, the above bound was achieved in a recent work of Alrabiah, Guruswami, Kothari and Manohar using an argument that crucially exploits known exponential lower bounds for $2$-LDCs. Their strategy hits an inherent bottleneck for $q \geq 5$.
Our key insight is identifying a general sufficient condition on the hypergraph of local decoding sets called $t$-approximate strong regularity. This condition demands that 1) the number of hyperedges containing any given subset of vertices of size $t$ (i.e., its co-degree) be equal to the same but arbitrary value $d_t$ up to a multiplicative constant slack, and 2) all other co-degrees be upper-bounded relative to $d_t$. This condition significantly generalizes related proposals in prior works that demand absolute upper bounds on all co-degrees.
We give an argument based on spectral bounds on Kikuchi Matrices that lower bounds the blocklength of any LDC whose local decoding sets satisfy $t$-approximate strong regularity for any $t \leq q$. Crucially, unlike prior works, our argument works despite having no non-trivial absolute upper bound on the co-degrees of any set of vertices. To apply our argument to arbitrary $q$-LDCs, we give a new, greedy, approximate strong regularity decomposition that shows that arbitrary, dense enough hypergraphs can be partitioned (up to a small error) into approximately strongly regular pieces satisfying the required relative bounds on the co-degrees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14361 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved Lower Bounds for all Odd-Query Locally Decodable Codes Basu, Arpon Hsieh, Jun-Ting Kothari, Pravesh K. Lin, Andrew D. Computational Complexity Combinatorics We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm1\}^k \rightarrow \{\pm1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in a sequence of prior works. For $q=3$, the above bound was achieved in a recent work of Alrabiah, Guruswami, Kothari and Manohar using an argument that crucially exploits known exponential lower bounds for $2$-LDCs. Their strategy hits an inherent bottleneck for $q \geq 5$. Our key insight is identifying a general sufficient condition on the hypergraph of local decoding sets called $t$-approximate strong regularity. This condition demands that 1) the number of hyperedges containing any given subset of vertices of size $t$ (i.e., its co-degree) be equal to the same but arbitrary value $d_t$ up to a multiplicative constant slack, and 2) all other co-degrees be upper-bounded relative to $d_t$. This condition significantly generalizes related proposals in prior works that demand absolute upper bounds on all co-degrees. We give an argument based on spectral bounds on Kikuchi Matrices that lower bounds the blocklength of any LDC whose local decoding sets satisfy $t$-approximate strong regularity for any $t \leq q$. Crucially, unlike prior works, our argument works despite having no non-trivial absolute upper bound on the co-degrees of any set of vertices. To apply our argument to arbitrary $q$-LDCs, we give a new, greedy, approximate strong regularity decomposition that shows that arbitrary, dense enough hypergraphs can be partitioned (up to a small error) into approximately strongly regular pieces satisfying the required relative bounds on the co-degrees. |
| title | Improved Lower Bounds for all Odd-Query Locally Decodable Codes |
| topic | Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2411.14361 |