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Quadratic residue code of length 42
[42, 21, 10]
Extended binary quadratic-residue code [42, 21, 10]. Weight-10 supports hold a 3-(42, 10, 18) design. Aut ≅ PSL(2, 41).
Forty-two seats in a circle. You choose ten. The rule is so strict that any three seats you point to sit together in exactly eighteen of those choices. The length is not a programmer’s joke. It is the extended quadratic-residue code of the prime 41. Pattern inside the code makes another pattern: a 3-design.
A code. Then a design.
[42, 21, 10]
3-(42, 10, 18)
What it supports
A named linear code whose length is 42 by construction (p + 1 for p = 41). Bonnecaze and Solé, Journal of Combinatorial Designs 2021: the 1722 weight-10 codewords hold a 3-(42, 10, 18) design. Automorphism group isomorphic to PSL(2, 41). The design is not given by Assmus-Mattson, and PSL(2, 41) is not 3-homogeneous, so transitivity does not explain it either. Magma computation, then a paper.
What it does not
Not ASCII 42. Not TIFF tag 42. Not a length chosen for Adams. The Coxeter graph’s 42 edges is 28 × 3 / 2, a different object. A unit-distance graph with 42 edges is not this code.
Calculation
[42, 21, 10] extended binary QR. wt-10 supports: 3-(42, 10, 18). Aut ≅ PSL(2, 41).
Quadratic-residue construction at the prime 41, then one bit of extension. Design by Magma count of triples, not Assmus-Mattson.
ExactnessExact
Form42
Unitlength
Strengthhard
When2021
Hung12 September 2026