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Yang's conjecture on near-normal sequences

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A base sequence in BS(n+1,n)BS(n+1,n) is a quadruple (A;B;C;D)(A;B;C;D) of {±1}\{\pm1\}-sequences with AA and BB of length n+1n+1, CC and DD of length nn, satisfying the base-sequence conditions; NN(n)NN(n) denotes the set of near-normal sequences, namely those satisfying bi=(−1)i−1aib_i=(-1)^{i-1}a_i for 1≤i≤n1\le i\le n. The existence of a near-normal sequence with n>1n>1 requires nn to be even.

Yang's conjecture. NN(n)≠∅NN(n)\ne\emptyset for all positive even integers nn.

Near-normal sequences are special base sequences used in constructions of orthogonal designs and Hadamard matrices. They are known to exist for every even n≤30n\le 30, while the conjecture concerns all positive even nn.

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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Yang's conjecture on near-normal sequences

    Let BS(n+1,n)BS(n+1,n) be a base sequence, and call (A,B,C,D)∈BS(n+1,n)(A,B,C,D)\in BS(n+1,n) a near-normal sequence, denoted NNS(n)NNS(n), when nn is even and

    bi=(−1)i−1ai(1≤i≤n),an+1=1,bn+1=−1.b_i=(-1)^{i-1}a_i\quad (1\le i\le n),\qquad a_{n+1}=1,\quad b_{n+1}=-1.

    Yang's conjecture. There is an NNS(n)NNS(n) for each even integer nn. The conjecture was verified for n≤40n\le40, but exhaustive search found no NNS(n)NNS(n) for n=42n=42 and n=44n=44, giving counterexamples.

    source: Xu Wang and Jiayi Zhu, “On Base, Normal and Near-normal Sequences”, arXiv:2506.20296 (2026).

References

Primary source

Dragomir Z. Djokovic, “Some new near-normal sequences”, arXiv:0907.3129 (2010).

combinatorial-designsdesign-theoryhadamard-matricesnear-normal-sequencessequences
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