Coding-Theory-Proof-Notes

Yang Shifu's Coding-Theory Notebook

Hello from Yang Shifu, a long-time researcher and enthusiastic traveller in sequence coding, coding theory, and finite-field combinatorics.

Yang Shifu · ORCID 0009-0002-2768-8506

National Mobile Communications Research Laboratory and the Frontiers Science Center for Mobile Information Communication and Security, Southeast University, Nanjing 211189, China

An expanding collection of mathematical proof notes on coding theory and sequence coding. Each note carries its own version and release date.

  1. The Exact Asymptotic Constant for Abelian Automorphism Groups over Finite Fields

    abelian automorphism groups; algebraic-geometry codes; conductors; finite fields; genus bounds.

    阿贝尔自同构群；代数几何码；导子；有限域；亏格界

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  2. Binary Cyclic Codes with Simultaneously Near-Linear Minimum and Dual Distances

    binary cyclic codes; minimum distance; dual distance; self-orthogonal codes; cyclic rank; entropy method; probabilistic existence.

    二元循环码；最小距离；对偶距离；自正交码；循环秩；熵方法；概率存在性

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  3. Nonexistence of Perfect Two-Error-Correcting Lee Codes

    Lee metric; perfect codes; Golomb--Welch conjecture; two-error correction; nonlinear codes; nonperiodic codes.

    Lee度量；完美码；Golomb--Welch猜想；二错误纠正；非线性码；非周期码

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  4. Global Optimizers of the Binary Masked Mixed-Qubit Bound

    binary codes; coding rate bounds; masked mixed-qubit channel; global optimization; interval arithmetic.

    二元码；码率上界；混合量子比特信道；全局优化；区间算术

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  5. Rank-Support Lifting and the Nonexistence of [8,3]64/2 Intersecting Codes

    rank-metric intersecting codes; rank support; binary finite fields; subspace intersection; length-eight codes.

    秩度量交码；秩支撑；二元有限域；子空间交；长度为八的码

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  6. The Universal Length-Eight Gap for Rank-Metric Intersecting Codes

    rank-metric intersecting codes; length-eight codes; finite fields; finite geometry; Hermitian geometry.

    秩度量交码；长度为八的码；有限域；有限几何；厄米特几何

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  7. Exact Large-Alphabet Capacity of Aperiodic Frequency-Hopping Sequence Sets

    aperiodic frequency-hopping sequences; Hamming correlation; sequence-set capacity; large alphabets; hypergraph matching.

    非周期跳频序列集；汉明相关；序列集容量；大字母表；超图匹配

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  8. The Euclidean-Weight GCD of Self-Dual Codes over Odd Residue Rings

    self-dual codes; Euclidean weights; odd residue rings; greatest common divisor; nonfree codes.

    自对偶码；欧几里得重量；奇数剩余类环；最大公约数；非自由码

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  9. The Covering-Radius Formula for Projective Reed-Solomon Codes

    projective Reed-Solomon codes; covering radius; MDS codes; normal rational curves; finite geometry; deep holes.

    射影里德所罗门码；覆盖半径；最大距离可分码；正规有理曲线；有限几何；深洞

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  10. Exact Half-Rate Points of the q-Ary Delsarte Bound

    q-ary codes; Delsarte linear programming bound; half-rate distance; Krawtchouk polynomials; auxiliary matrices; Potts model.

    q元码；Delsarte线性规划界；半码率距离；Krawtchouk多项式；辅助矩阵；Potts模型

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  11. Strict Nonbinary Rate Bounds Beyond Optimized Classical and Mixed-Channel Envelopes

    nonbinary codes; asymptotic coding rate bounds; Aaltonen bound; mixed-channel bounds; Plotkin endpoint; classical-quantum channels.

    非二元码；渐近码率上界；Aaltonen界；混合信道界；Plotkin端点；经典量子信道

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