Nearly Tight Lower Bounds for Relaxed Locally Decodable Codes via Robust Daisies
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| Format: | Preprint |
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2025
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| _version_ | 1866918218973577216 |
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| author | Goldberg, Guy Gur, Tom Saraogi, Sidhant |
| author_facet | Goldberg, Guy Gur, Tom Saraogi, Sidhant |
| contents | We show a nearly optimal lower bound on the length of linear relaxed locally decodable codes (RLDCs). Specifically, we prove that any $q$-query linear RLDC $C\colon \{0,1\}^k \to \{0,1\}^n$ must satisfy $n = k^{1+Ω(1/q)}$. This bound closely matches the known upper bound of $n = k^{1+O(1/q)}$ by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004).
Our proof introduces the notion of robust daisies, which are relaxed sunflowers with pseudorandom structure, and leverages a new spread lemma to extract dense robust daisies from arbitrary distributions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_21659 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nearly Tight Lower Bounds for Relaxed Locally Decodable Codes via Robust Daisies Goldberg, Guy Gur, Tom Saraogi, Sidhant Computational Complexity Information Theory Combinatorics 94B65 (Primary) 05D05, 94B35, 68Q17 (Secondary) E.4; F.2.2; G.2.1 We show a nearly optimal lower bound on the length of linear relaxed locally decodable codes (RLDCs). Specifically, we prove that any $q$-query linear RLDC $C\colon \{0,1\}^k \to \{0,1\}^n$ must satisfy $n = k^{1+Ω(1/q)}$. This bound closely matches the known upper bound of $n = k^{1+O(1/q)}$ by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004). Our proof introduces the notion of robust daisies, which are relaxed sunflowers with pseudorandom structure, and leverages a new spread lemma to extract dense robust daisies from arbitrary distributions. |
| title | Nearly Tight Lower Bounds for Relaxed Locally Decodable Codes via Robust Daisies |
| topic | Computational Complexity Information Theory Combinatorics 94B65 (Primary) 05D05, 94B35, 68Q17 (Secondary) E.4; F.2.2; G.2.1 |
| url | https://arxiv.org/abs/2511.21659 |