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1. LOCO Codes: Lexicographically-Ordered Constrained Codes

   https://doi.org/10.1109/TIT.2019.2943244  

   Hareedy, Ahmed ; Calderbank, Robert ( June 2020 , IEEE Transactions on Information Theory)
2. Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes

   https://doi.org/10.1007/s10623-020-00726-x  

   López, Hiram H. ; Matthews, Gretchen L. ; Soprunov, Ivan ( January 2020 , Designs, Codes and Cryptography)
3. Asymmetric LOCO Codes: Constrained Codes for Flash Memories

   https://doi.org/10.1109/ALLERTON.2019.8919932  

   Hareedy, Ahmed ; Calderbank, Robert ( September 2019 , 57th Annual Allerton Conference on Communications, Control and Computing)

   In data storage and data transmission, certain patterns are more likely to be subject to error when written (transmitted) onto the media. In magnetic recording systems with binary data and bipolar non-return-to-zero signaling, patterns that have insufficient separation between consecutive transitions exacerbate inter-symbol interference. Constrained codes are used to eliminate such error-prone patterns. A recent example is a new family of capacity-achieving constrained codes, named lexicographically-ordered constrained codes (LOCO codes). LOCO codes are symmetric, that is, the set of forbidden patterns is closed under taking pattern complements. LOCO codes are suboptimal in terms of rate when used in Flash devices where block erasure is employed since the complement of an error-prone pattern is not detrimental in these devices. This paper introduces asymmetric LOCO codes (A-LOCO codes), which are lexicographically-ordered constrained codes that forbid only those patterns that are detrimental for Flash performance. A-LOCO codes are also capacity-achieving, and at finite-lengths, they offer higher rates than the available state-of-the-art constrained codes designed for the same goal. The mapping-demapping between the index and the codeword in A-LOCO codes allows low-complexity encoding and decoding algorithms that are simpler than their LOCO counterparts. 

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4. Graph-codes

   https://doi.org/10.1016/j.ejc.2023.103880  

   Alon, Noga ( February 2024 , European Journal of Combinatorics)
5. Antichain codes

   https://doi.org/10.1112/blms.12909  

   Gunby, Benjamin ; He, Xiaoyu ; Narayanan, Bhargav ; Spiro, Sam ( August 2023 , Bulletin of the London Mathematical Society)

   A family of sets is said to be an antichain if for all distinct , and it is said to be a distance‐ code if every pair of distinct elements of has Hamming distance at least . Here, we prove that if is both an antichain and a distance‐ code, then . This result, which is best‐possible up to the implied constant, is a purely combinatorial strengthening of a number of results in Littlewood–Offord theory; for example, our result gives a short combinatorial proof of Hálasz's theorem, while all previously known proofs of this result are Fourier‐analytic.

    

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