$m$-Sequences of Different Lengths with Four-Valued Cross Correlation

{\bf Abstract.} Considered is the distribution of the cross correlation between $m$-sequences of length $2^m-1$, where $m$ is even, and $m$-sequences of shorter length $2^{m/2}-1$. The infinite family of pairs of $m$-sequences with four-valued cross correlation is constructed and the complete correlation distribution of this family is determined.

💡 Research Summary

The paper investigates the cross‑correlation distribution between binary maximal‑length sequences (m‑sequences) of two different lengths: a long sequence of period (2^{m}-1) and a short sequence of period (2^{m/2}-1), where the exponent (m) is an even integer. While the correlation properties of m‑sequences of equal length have been extensively studied—yielding families such as Gold (two‑valued) and Kasami (three‑valued)—much less is known when the two sequences have different periods. This work fills that gap by constructing an infinite family of pairs of m‑sequences whose cross‑correlation takes exactly four distinct values, and by providing the complete frequency distribution of those values.

Construction of the sequences.
Let (\mathbb{F}{2^{m}}) and (\mathbb{F}{2^{m/2}}) be the finite fields of orders (2^{m}) and (2^{m/2}), respectively. Choose primitive elements (\alpha\in\mathbb{F}{2^{m}}) and (\beta\in\mathbb{F}{2^{m/2}}). The long m‑sequence (S={s_i}) and the short m‑sequence (T={t_j}) are defined by the trace representations
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