New results on nonexistence of perfect p-ary sequences and almost p-ary sequences

On the Non-Existence of Certain Classes of Perfect $p$ -Ary Sequences and Perfect Almost $p$ -Ary Sequences

“…Proposition 5.1 (See [12] Theorem 1.4 (2)). If there exist PAPS with type [p, n + 1], then p | n − 1 and αᾱ = n for some α ∈ Z[ζ p ].…”

“…It follows that q is not self-conjugate w.r.t p, which says that our case is not contained in [14] ((1) in the above list). Moreover, our case is also different from [12] ((2) in the above list) since p ≡ 3 (mod 4). Thus our result is new.…”

“…Then ord p (q) = (p − 1)/4 > 1 and so (p, q − 1) = 1. Recall that E p is the decomposition field of q in K, so we use [12,Lemma 2.4 (3)] to obtain that ββ = q for some β ∈ o K and β 2 ∈ o Ep .…”

“…Their results imply that PPSs (PAPSs) can be constructed if the corresponding relative difference sets exist. By using various techniques in combinatorial design theory, several classes of such sequences have been constructed (see [2,9,12,10]). On the other hand, there are some nonexistence results on such sequences (and related difference sets), see [2,9,14,17].…”

“…Remark 1.6. For the same reason, this case is also not contained in [14] ((1) in the above list), and the result [12] ((2) in the above list) can only deal with type [p, p a q l n ′ ] where l < λ/s. By direct calculation for (p, f ) in the cases listed in Theorem 1.5, the corresponding λ/s ≤ l 0 .…”