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arXiv:math/0703655v2

On an Improvement of a Result by Niederreiter and Wang Concerning the Expected Linear Complexity of Multisequences

Nikolai Moshchevitin, Michael Vielhaber

math.NT11B85
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Paper metadata

Original submission
March 22, 2007
Latest revision
October 20, 2007 · version 2
Claimed MSC
11B85
arXiv categories
math.NT
Comments
5 pages, minor corrections of the proofs

Abstract

We show that the expected value for the linear complexity of mm-multisequences of length nn is E_n^{(m)} = n m/(m+1) + O(1).

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 20, 2026
01Statements1 reported findingCorrect

The expected joint linear complexity is mn/(m+1)+O(1)mn/(m+1)+O(1) for every fixed number of component sequences.

Main theoremCorrect

Uniformly bounded expected-complexity deviation

Pages 2–5 · main theorem and proof · arXiv:math/0703655v2

The two cited counting estimates give exponential decay in the absolute deviation from mn/(m+1)mn/(m+1). Summing the deviation against that decay is uniformly bounded in nn, which yields the stated expectation after the harmless rounding by a ceiling.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The main summation is correct, but the proof of the auxiliary lattice-point bound contains a false cube inclusion. A verified coordinate-range argument proves the same bound.

Tail summationCorrect

The two deviation tails give a uniform expectation bound

Pages 4–5 · final proof of the main theorem · arXiv:math/0703655v2

Splitting at L=⌈mn/(m+1)⌉L=\lceil mn/(m+1)\rceil, the cited lower- and upper-tail counts bound the probability of a deviation of size hh by a summable geometric sequence with only a polynomial factor. Summing hh against those tails is bounded independently of nn, which proves the asserted O(1)O(1) expectation error.

Polytope lattice-point lemmaIncorrect as written · verified repair

The simplex is not contained in the printed one-sided cube

Page 4 · proof of MH≤(H+1)mM_H\leq(H+1)^m · arXiv:math/0703655v2

The displayed vertex coordinates include values L/m+H/(mν)L/m+H/(m\nu), so ΩH∗\Omega_H^* is not contained in [L/m−H,L/m]m[L/m-H,L/m]^m. For a fixed coordinate jj, however, its maximum and minimum over all vertices differ by at most Hm(1j+1m−j+1)≤H.\frac{H}{m}\left(\frac1j+\frac1{m-j+1}\right)\leq H. Hence every coordinate lies in some interval of length at most HH, containing at most H+1H+1 integers. Taking the product over the mm coordinates gives the claimed MH≤(H+1)mM_H\leq(H+1)^m, so all subsequent exponential estimates remain valid.

Expected-value notationTypo

The multisequence space has an extra pair of parentheses

Page 2 · definition of En(m)E_n^{(m)} · arXiv:math/0703655v2

Replace (Fq(m))n(F_q^{(m)})^n by (Fqm)n(F_q^m)^n. The multisequence TT is defined with that ambient space in the same sentence.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:math/0703655v2
Authors listed
Nikolai Moshchevitin, Michael Vielhaber
Audit date
August 20, 2026
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