data transmission engineers

Abstract: Each of the 64 hexagrams of the I-Ching, or Book of Changes, is a stack of six Yin-Yang lines and is therefore a binary vector of length 6; the eight hexagrams formed by repeating a single trigram span a doubled-trigram subspace $Δ$. We ask which binary $[6,3,3]$ codes are compatible with this trigram structure. We show that a $[6,3,3]$ code meets $Δ$ in at most four codewords, that the four are then forced to be the doubled trigrams Earth, Fire, Wind and Lake, and that exactly two codes attain this bound: the shortened Hamming code $\mathcal C_A$ and its dual. Since $\mathcal C_A$ has covering radius 2, its eight codewords, their radius-one balls, and its eight deep holes together reconstruct all 64 hexagrams. In general, we call a binary code of length $2m$ with $d \ge 3$ diagonal-coherent if it contains the half of the diagonal subspace $\{(u,u)\}$ coming from even-weight vectors $u$ -- the largest part of that subspace a single-error-correcting code can contain -- and extremal if its dimension is as large as possible. For $m\ge2$ and $r=\lceil\log_2(2m+1)\rceil$, we determine the number $N(m)$ of extremal diagonal-coherent codes with parameters $[\,2m,\ 2m-r,\ \ge3\,]$, show that diagonal coherence entails no loss of redundancy, and determine their minimum distances. We also put these codes in a normal form, determine their hulls, and show that each has covering radius $2$ with exactly $2^r-1-2m$ deep-hole cosets. Finally we propose three open problems.

Abstract: Let $u$ be a received word for the full-length Reed--Solomon code $\operatorname{RS}_q(k)$, with interpolation polynomial $f_u$ of degree $k+m$. For each $m\in\{3,4,5,6,7\}$, we classify all such deep holes over every finite field, without a field-size hypothesis. None occur for odd $m$ in this range. At $m=4$ and $m=6$, the only exceptions are, respectively, the Segre sextic and Frobenius-power families; these occur only in dimension two over fields of characteristic two. For arbitrary $m\geqslant3$, we also establish two uniform bounds. If $\sqrt q\geqslant12(m+1)\log(2m+2)$, no word of interpolation degree $k+m$ is a deep hole when $q$ is odd, $k\geqslant3$, or $m$ is odd. If $q>16(m+2)^4$, we obtain the corresponding complete classification at every admissible $k$, including the explicit dimension-two exceptions. Our proofs use an agreement criterion expressed in complete homogeneous symmetric functions, agreement-count estimates, finite geometry, permutation-polynomial restrictions, and subset constructions for solving the resulting symmetric-function equations. The remaining finite cases are settled by finite computations with supplied certificates and deterministic verification programs.

Abstract: We introduce matrix $Φ$-ribbons, extending the scalar $Φ$-ribbons to the matrix case. For $Φ$ in the subadditivity class, we establish data processing and tensorization properties of matrix $Φ$-ribbons. We establish scalar and matrix $Φ$-entropy inequalities and use them to show that both $Φ$-ribbons characterize the corresponding $Φ$-strong data processing inequality (SDPI) constants. We obtain exact evaluations of the SDPI constants for several sources. We provide examples showing that the matrix $Φ$-SDPI constant can exceed the scalar counterpart. Finally, we derive inner bounds for $Φ$-ribbons from prescribed independence structures and prove that these bounds are optimal when $Φ(t)=t\log t$.

Abstract: Evolutionary program search driven by large language models (LLMs) has produced record-breaking constructions for open problems in combinatorics and beyond. We apply this approach to the longstanding problem of improving the best-known bounds for binary linear codes. Building on the EvoTune evolutionary framework and the ShinkaEvolve codebase, we introduce LinCodeEvolve, which evolves code-construction programs against an exact minimum-distance evaluator. A strategy loop combines diversity-driven search and expert supervision: when progress plateaus, new strategies are used to redirect the search. LinCodeEvolve discovers seven record-breaking codes, $[172,21,66]$, $[173,20,68]$, $[176,21,68]$, $[181,21,70]$, $[184,21,72]$, $[189,22,72]$ and $[200,21,77]$, six of which have concise quasi-cyclic descriptions. With standard code modification techniques, they improve $22$ entries of the tables. Every code is verified by exhaustive enumeration. These results suggest that LLM-guided search can help find improved codes and complement existing methods in coding theory.