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arXiv 2506.18878cs.ITcs.CCmath.IT
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遗忘删除码

Oblivious Deletion Codes

Roni Con, Ray Li

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AI总结：

本文在遗忘错误模型下构造删除纠错码，通过素数模哈希减小冗余，实现显式与随机化构造，分别匹配或优于对抗性删除的存在性冗余界。

摘要：

我们在遗忘模型中构造删除纠错码，其中错误是对抗性的，但对编码器的随机性不知情。遗忘错误弥合了对抗性错误模型与随机错误模型之间的差距，其动机源于DNA存储等应用，在这些应用中，噪声由难以建模的物理现象引起，而非由对手造成。\n(1)（显式遗忘）我们构造了$t$个遗忘删除码，冗余度约为$2t\log n$，匹配对抗性删除的存在性界。\n(2)（列表解码蕴含显式遗忘）我们证明显式列表可解码码能产生参数基本相同的显式遗忘码。根据Guruswami和Håstad的工作（IEEE TIT, 2021），这给出了冗余度约为$3\log n$的2个遗忘删除码，优于2个对抗性删除的存在性冗余。\n(3)（随机化遗忘）我们给出遗忘码的随机化构造，以至少$1-2^{-n}$的概率产生一个能纠正$t$个遗忘删除的码，冗余度约为$(t+1)\log n$，优于对抗性删除的存在性冗余$\sim 2t\log n$。\n(4)（随机化对抗）研究遗忘模型可为对抗性码的更好构造提供信息。同样的技术以至少$1-2^{-n}$的概率产生一个能纠正$t$个对抗性删除的码，冗余度约为$(2t+1)\log n$，几乎匹配存在性冗余$\sim 2t\log n$。\n这些结果背后的共同思想是：通过从小子集中（随机）选择一个素数进行模运算来减小哈希大小，并在哈希中包含该素数的短编码。

英文摘要：

We construct deletion error-correcting codes in the oblivious model, where errors are adversarial but oblivious to the encoder's randomness. Oblivious errors bridge the gap between the adversarial and random error models, and are motivated by applications like DNA storage, where the noise is caused by hard-to-model physical phenomena, but not by an adversary. (1) (Explicit oblivious) We construct $t$ oblivious deletion codes, with redundancy $\sim 2t\log n$, matching the existential bound for adversarial deletions. (2) (List decoding implies explicit oblivious) We show that explicit list-decodable codes yield explicit oblivious deletion codes with essentially the same parameters. By a work of Guruswami and Håstad (IEEE TIT, 2021), this gives 2 oblivious deletion codes with redundancy $\sim 3\log n$, beating the existential redundancy for 2 adversarial deletions. (3) (Randomized oblivious) We give a randomized construction of oblivious codes that, with probability at least $1-2^{-n}$, produces a code correcting $t$ oblivious deletions with redundancy $\sim(t+1)\log n$, beating the existential adversarial redundancy of $\sim 2t\log n$. (4) (Randomized adversarial) Studying the oblivious model can inform better constructions of adversarial codes. The same technique produces, with probability at least $1-2^{-n}$, a code correcting $t$ adversarial deletions with redundancy $\sim (2t+1)\log n$, nearly matching the existential redundancy of $\sim 2t\log n$. The common idea behind these results is to reduce the hash size by modding by a prime chosen (randomly) from a small subset, and including a small encoding of the prime in the hash.

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