More Like this

1. Time and Space Efficient Deterministic List Decoding

   https://doi.org/10.4230/lipics.itcs.2026.42  

   Cook, Joshua; Moshkovitz, Dana (January 2026, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Saraf, Shubhangi (Ed.)

   Error correcting codes encode messages by codewords in such a way that even if some of the codeword is corrupted, the message can be decoded. Typical decoding algorithms for error correcting codes either use linear space or quadratic time. A natural question is whether codes can be decoded in near-linear time and sub-linear space simultaneously. A recent result by Cook and Moshkovitz gave efficient decoders that can uniquely decode Reed-Muller and other codes from a constant fraction (less than half) of corruption. In this work, we address the problem of list decoding in near-linear time and sub-linear space. In the list decoding setting, most of the codeword is corrupted, and one wants to output a short list of potential messages that contains the true message. For any constants γ, τ > 0, we give decoders for Reed-Muller codes that can decode from 1-γ fraction of corruptions in time n^{1+τ} and space n^{τ}. Our decoders work by extending the iterative correction technique of Cook and Moshkovitz. However, that technique, which gradually decreases the number of corruptions in the message, was tailored to the unique decoding setting. We first identify an intermediate problem, codewords list recovery, for which we can make iterative correction work. We then show how to reduce general list decoding to the codewords list recovery problem in efficient time and space. The reduction relies on local correction and testing. In the codewords list recovery problem, the input consists of n unordered lists containing exactly the symbols from L codewords, where a small fraction of the lists is corrupted. The goal is to find the L codewords. In addition, we prove that any linear code with time-space efficient encoding or decoding must be local, in the sense that the codewords satisfy a local linear constraint. This rules out codes like Reed-Solomon from having time-space efficient encoding or decoding. 

   more » « less
2. ELF Codes: Concatenated Codes with an Expurgating Linear Function as the Outer Code

   https://doi.org/10.1109/ISTC57237.2023.10273535  

   Wesel, R.; Antonini, A.; Wang, L.; Sui, W.; Towell, B.; Grissett, H. (September 2023, IEEE)

   An expurgating linear function (ELF) is an outer code that disallows low-weight codewords of the inner code. ELFs can be designed either to maximize the minimum distance or to minimize the codeword error rate (CER) of the expurgated code. A list-decoding sieve can efficiently identify ELFs that maximize the minimum distance of the expurgated code. For convolutional inner codes, this paper provides analytical distance spectrum union (DSU) bounds on the CER of the concatenated code. For short codeword lengths, ELFs transform a good inner code into a great concatenated code. For a constant message size of K = 64 bits or constant codeword blocklength of N = 152 bits, an ELF can reduce the gap at CER 10−6 between the DSU and the random-coding union (RCU) bounds from over 1 dB for the inner code alone to 0.23 dB for the concatenated code. The DSU bounds can also characterize puncturing that mitigates the rate overhead of the ELF while maintaining the DSU-to-RCU gap. List Viterbi decoding guided by the ELF achieves maximum likelihood (ML) decoding of the concatenated code with a sufficiently large list size. The rate-K/(K+m) ELF outer code reduces rate and list decoding increases decoder complexity. As SNR increases, the average list size converges to 1 and average complexity is similar to Viterbi decoding on the trellis of the inner code. For rare large-magnitude noise events, which occur less often than the FER of the inner code, a deep search in the list finds the ML codeword. 

   more » « less
3. On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors

   https://doi.org/10.4230/LIPIcs.CCC.2023.14  

   Block, Alexander R.; Blocki, Jeremiah; Cheng, Kuan; Grigorescu, Elena; Li, Xin; Zheng, Yu; Zhu, Minshen (January 2023, 38th Computational Complexity Conference (CCC 2023))

   Ta-Shma, Amnon (Ed.)

   Locally Decodable Codes (LDCs) are error-correcting codes C:Σⁿ → Σ^m, encoding messages in Σⁿ to codewords in Σ^m, with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length m that is super-polynomial in n, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (SICOMP 2006) show how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs. In this work, we study RLDCs in the standard Hamming-error setting, and introduce their variants in the insertion and deletion (Insdel) error setting. Standard LDCs for Insdel errors were first studied by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), and are further motivated by recent advances in DNA random access bio-technologies. Our first result is an exponential lower bound on the length of Hamming RLDCs making 2 queries (even adaptively), over the binary alphabet. This answers a question explicitly raised by Gur and Lachish (SICOMP 2021) and is the first exponential lower bound for RLDCs. Combined with the results of Ben-Sasson et al., our result exhibits a "phase-transition"-type behavior on the codeword length for some constant-query complexity. We achieve these lower bounds via a transformation of RLDCs to standard Hamming LDCs, using a careful analysis of restrictions of message bits that fix codeword bits. We further define two variants of RLDCs in the Insdel-error setting, a weak and a strong version. On the one hand, we construct weak Insdel RLDCs with almost linear codeword length and constant query complexity, matching the parameters of the Hamming variants. On the other hand, we prove exponential lower bounds for strong Insdel RLDCs. These results demonstrate that, while these variants are equivalent in the Hamming setting, they are significantly different in the insdel setting. Our results also prove a strict separation between Hamming RLDCs and Insdel RLDCs. 

   more » « less
4. Efficient Nested Key Equation Solver for Short Generalized Integrated Interleaved BCH Codes

   Zhenshan Xie and Xinmiao Zhang (May 2022, Proceedings IEEE International Symposium on Circuits and Systems)

   Generalized integrated interleaved (GII) codes can nest BCH sub-codewords to form stronger BCH codewords. They are among the best candidates for error correction in the new storage class memories (SCMs). However, SCMs require short codeword length and low redundancy. In this case, the nested key equation solver (KES), which is a key step in GII decoding, has a small number of iterations. The initialization and/or scalar pre-computation in previous nested KES designs have large area and may take even longer time than the iterations themselves. This paper proposes an efficient nested KES design for short GII-BCH codes. The polynomial updating is decomposed into two steps to reduce the critical path without requiring scalar pre-computation. Besides, the KES is reformulated to reduce the number of clock cycles without incurring any area overhead. For an example code over GF(2^{10}) that protects 2560 bits with 10% redundancy, the proposed design achieves at least 25% area reduction and 37% reduction on the area-time product averaged over the nested decoding rounds compared to prior efforts. 

   more » « less
5. CRC-Aided List Decoding of Convolutional and Polar Codes for Short Messages in 5G

   https://doi.org/10.1109/ICC45855.2022.9838726  

   King, J.; Kwon, A.; Yang, H.; Ryan, W.; Wesel, R. D. (May 2022, 2022 International Conference on Communications)

   This paper explores list decoding of convolutional and polar codes for short messages such as those found in the 5G physical broadcast channel. A cyclic redundancy check (CRC) is used to select a codeword from a list of likely codewords. One example in the 5G standard encodes a 32-bit message with a 24-bit CRC and a 512-bit polar code with additional bits added by repetition to achieve a very low rate of 32/864. This paper shows that optimizing the CRC length improves the Eb/N0 performance of this polar code, where Eb/N0 is the ratio of the energy per data bit to the noise power spectral density. Furthermore, even better Eb/N0 performance is achieved by replacing the polar code with a tail-biting convolutional code (TBCC) with a distance-spectrum-optimal (DSO) CRC. This paper identifies the optimal CRC length to minimize the frame error rate (FER) of a rate-1/5 TBCC at a specific value of Eb/N0. We also show that this optimized TBCC/CRC can attain the same excellent Eb/N0 performance with the very low rate of 32/864 of the 5G polar code, where the low rate is achieved through repetition. We show that the proposed TBCC/CRC concatenated code outperforms the PBCH polar code described in the 5G standard both in terms of FER and decoding run time. We also explore the tradeoff between undetected error rate and erasure rate as the CRC size varies. 

   more » « less