Ordered Orthogonal Array Construction Using LFSR Sequences

A General Construction of Ordered Orthogonal Arrays Using LFSRs

“…The next three lemmas prove properties of P l,r . These results were shown as Theorem 1 in [2], but the proofs given here are significantly shorter and less complicated than the ones given there. These lemmas are used in Section 3 to prove the correctness of the construction given in this paper.…”

“…when s 0 = 0), then r has a run of l zeroes at index 0. Momentarily disregarding the fourth assumption that r has a run of l zeroes at index 0, we note that even if s 0 = 0, Lemma 2 shows that it is still possible to obtain a run of l zeroes in r. In [2], a special polynomial was constructed from r whose roots determine the exact conditions under which the run of l zeroes in r came from a run of l + 1 zeroes in s. This polynomial is:…”

“…Equivalently, γ is the number of β ∈ F q that are not roots of f . When f is primitive, γ = q and the resulting OOA(q t ; t, q + 1, t, q) is essentially the one produced in [2].…”

“…By Theorem 9, this is an OOA (16; 4, 3, 4, 2) and is essentially the one that would be constructed by the RUNS construction of [2].…”

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In [2], q t × (q + 1)t ordered orthogonal arrays (OOAs) of strength t over the alphabet F q were constructed using linear feedback shift register sequences (LFSRs) defined by primitive polynomials in F q [x]. In this paper we extend this result to all polynomials in F q [x] which satisfy some fairly simple restrictions, restrictions that are automatically satisfied by primitive polynomials.

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