[Paper Review] Some new near-normal sequences

Dragomir Ž. Djoković|arXiv (Cornell University)|Jul 17, 2009

Approximation Theory and Sequence Spaces4 references3 citations

TL;DR

This paper establishes the existence of near-normal sequences for n = 32 and n = 34, confirming Yang's conjecture for these even values. Using exhaustive computer searches, the authors classify 8 NN-equivalence classes for n = 32 and construct two non-equivalent classes for n = 34, while proving NS(31) and NS(33) are empty, thereby showing 63 and 67 are not Yang numbers, and 69 is a new Yang number.

ABSTRACT

The normal sequences NS(n) and near-normal sequences NN(n) play an important role in the construction of orthogonal designs and Hadamard matrices. They can be identified with certain base sequences (A;B;C;D), where A and B have length n+1 and C and D length n. C.H. Yang conjectured that near-normal sequences exist for all even n. While this has been confirmed for n not exceeding 30, so far nothing else was known for larger n. We show that NN(32) consists of 8 equivalence classes and we exhibit their representatives. We also construct representatives for two equivalence classes of NN(34). On the other hand our exhaustive computer searches show that NS(31) and NS(33) are void.

Motivation & Objective

• To investigate the existence of near-normal sequences (NN(n)) for even n > 30, extending prior results up to n=30.
• To confirm or refute C.H. Yang's conjecture that NN(n) is non-empty for all even n.
• To determine whether normal sequences (NS(n)) exist for odd n, particularly n=31 and n=33, as these are critical for identifying Yang numbers.
• To classify the structure of NN(32) and construct representatives for NN(34) using computational enumeration under NN-equivalence.
• To establish new Yang numbers by analyzing the non-emptiness of NS(n) and NN(n), with implications for Hadamard matrix constructions.

Proposed method

• Employed exhaustive computer searches to enumerate all near-normal sequences in NN(n) under NN-equivalence for n=32 and partial enumeration for n=34.
• Used the same algorithm as in prior work [2] to generate and verify base sequences satisfying the norm condition N(A)+N(B)+N(C)+N(D)=2(2n+1) for n=32 and n=34.
• Verified that candidate sequences satisfy the near-normal condition: b_i = (-1)^{i-1} a_i for 1 ≤ i ≤ n.
• Represented sequences in a compact encoded form for efficient storage and comparison, with detailed encoding described in prior works [2,3].
• Applied the Goethals–Seidel array construction to generate Hadamard matrices from valid base sequences.
• Used the definition of Yang numbers (odd integers 2s+1 where NS(s) or NN(s) is non-empty) to infer the status of 63, 67, and 69.

Experimental results

Research questions

• RQ1 Does near-normal sequence existence hold for n=32 and n=34, extending Yang's conjecture beyond n=30?
• RQ2 Are there any normal sequences for n=31 and n=33, and what does this imply for the status of 63, 67, and 69 as Yang numbers?
• RQ3 How many NN-equivalence classes exist for NN(32), and what are their structural properties?
• RQ4 Can new non-equivalent near-normal sequences be constructed for n=34, and how do they compare to known classes?
• RQ5 What is the impact of these findings on the construction of Hadamard matrices via Yang multiplication and Williamson-type matrices?

Key findings

• NN(32) consists of exactly 8 NN-equivalence classes, with representatives explicitly constructed and encoded.
• Two non-equivalent near-normal sequences were constructed for NN(34), though the full classification remains incomplete.
• NS(31) and NS(33) are both empty, as confirmed by exhaustive computer search.
• 63 and 67 are not Yang numbers, since neither NS(31) nor NS(33) is non-empty.
• 69 is a new Yang number, as NN(34) is non-empty, and this leads to the construction of infinitely many new Hadamard matrices via Yang multiplication.
• The results confirm Yang's conjecture for n=32 and n=34, extending the known range of even n for which near-normal sequences exist.