S H N O U D I R E S E A R C H On the Putative Extremal [72, 36, 16] Binary Code

@misc{shnoudi2026s,
  author       = {Shnoudi, Ibrahim Amin and Shnoudi, Ibrahim Amin},
  title        = {{S H N O U D I R E S E A R C H On the Putative Extremal [72, 36, 16] Binary Code}},
  year         = {2026},
  month        = oct,
  date         = {2026-10-03},
  publisher    = {Life of Research},
  howpublished = {Preprint},
  url          = {https://lifeofresearch.com/preprints/80},
  abstract     = {Whether a binary doubly-even self-dual [72,36,16] code exists has been open since 1973. This document does not settle the question; it maps, with certified exactness, the point at which the known methods stop, so that a future argument can begin there. On the way it proves the results below about such a code, which were not found in the sources consulted. An explicit form of Delsarte's theorem on the weight distributions of cosets expresses the Walsh spectrum of the minimum words on every coset through the numbers of vectors of weight at most 10 in that coset. It gives a congruence: every hyperplane of an extremal code contains a number of minimum words congruent to 25 modulo 32 if it contains the all-ones word and to 23 modulo 128 otherwise, which excludes as hyperplanes the two [72,35,16] codes reported in [10]; it makes the deep odd cosets rigid; and it leaves every extremal code with more than two billion hyperplanes whose count is 124 567 or 124 695. Comparing the formula with the design expansion re-derives exact local identities; for instance every set of seven coordinates meets exactly 78 minimum words in at least six points, which is equivalent to the vanishing of the degree-7 harmonic component. Around these, an extension criterion reformulates existence as a finite, checkable condition on doubly-even self-orthogonal [72,35,16] codes containing the all-ones word: such a code extends to an extremal code precisely when the words of weight 4, 8 and 12 of its dual are pairwise orthogonal; it is an explicit form of the standard neighbour argument and does not by itself shorten the search. The document also derives the intersection distribution around a minimum word, glues these distributions at pairs of minimum words and shows that the glued systems are polytope-feasible and, at one pair, integer-feasible, so that layer cannot decide d₃ ∈ \{28, 29\}. It describes the local structure of λ₆, the first multiplicity the 5-design leaves undetermined, and shows that the first four power sums of the spectrum divide equally between the two kinds of hyperplanes. The constraints on the multiplicity function forced on the code admit an explicit non-negative integer solution satisfying every local identity on the support of a minimum word, so no contradiction is obtained from that layer either; this concerns the constraints, not a code. Every step that admits a control is checked at n = 24 and n = 48, where the extremal code exists, before n = 72; the same tools are run at length 96. Every computational statement is classified as equation-consistent, polytope-feasible, integer-feasible or code-realisable, and no implication from the first three levels to the existence of a code is asserted. All computations are certified in the appendices by self-contained scripts with the output of their runs. No claim of nonexistence is made.},
  keywords     = {Shnoudi Theory.. Mathematics..Physics},
  language     = {en},
  copyright    = {Creative Commons Attribution 4.0}
}