Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings. (English) Zbl 0941.94015
The paper is concerned with the construction of frequency hopping patterns suitable for frequency hopping multiple access communication systems. The constructed sequences are optimal in the sense that their Hamming cross-correlation function meets a bound that was derived by Lempel and Greenberger. One of the construction methods (leading to nonlinear sequences) gives in addition sequences with a large linear complexity. The constructed frequency hopping patterns are derived from sequences over a Galois ring \(R=F[\xi/(\omega(\xi))]^k\). In fact the constructed sequences are combinations of \(m\)-sequences over a finite fields and the \(\omega\) is used as a kind of book keeping mechanism to keep track (to separate) these \(m\)-sequences. The material in the paper is quite interesting but due to the presentation the paper is hard to read. In that respect I want to make the following comments.
The proof of Lemma 1 is a bit fishy and lacks some of the arguments. The intention is to arrive at the representation of \(R\) as \(F+\omega F+\omega^2 F+\dots + \omega^{k-1} F\) where \(F\) is the field \(F_{PRC}\cup 0\), which is something that becomes more clear by stating the fact and giving some examples.
In Example 3 the \(r-1\) in the formula equals 2 since \(r\) equals 3. Formula (16) is not an equation, but it should be since the authors define \(S(f)\) as the family of sequences satisfying (16). The sequences in Table III start with Tr\((\nu\alpha)\) instead of with Tr\((\nu)\) which is not according to the definition in Theorem 1.
The statements in Lemma 9 and 10 are only valid for \(k=r\), otherwise \((t,\rho)\) is the number of \(r\times k\) matrices over \(GF(p^m)\). This can easily be checked by using the results of table III in which there are only 2 sequences of rank 1 while the number of \(3\times 3\) matrices of rank 1 starting with column \((1 0 0)^T\) is 4. Table IV however remains valid since in that case \(r=k\).
In part B of Section IV \(\Psi\) is not defined, probably it is \(\Psi^1\). I do not understand how formula (24) is derived, and have some doubts on its correctness. The proofs of Lemma 13 and 14 are unclear. In Example 5 of Section E, \(x\to x^3\) is used as a permutation polynomial of the intermediate ring, but this is not stated. The sequence in Example 6 is shifted the same way as the sequence of table III.
The proof of Lemma 1 is a bit fishy and lacks some of the arguments. The intention is to arrive at the representation of \(R\) as \(F+\omega F+\omega^2 F+\dots + \omega^{k-1} F\) where \(F\) is the field \(F_{PRC}\cup 0\), which is something that becomes more clear by stating the fact and giving some examples.
In Example 3 the \(r-1\) in the formula equals 2 since \(r\) equals 3. Formula (16) is not an equation, but it should be since the authors define \(S(f)\) as the family of sequences satisfying (16). The sequences in Table III start with Tr\((\nu\alpha)\) instead of with Tr\((\nu)\) which is not according to the definition in Theorem 1.
The statements in Lemma 9 and 10 are only valid for \(k=r\), otherwise \((t,\rho)\) is the number of \(r\times k\) matrices over \(GF(p^m)\). This can easily be checked by using the results of table III in which there are only 2 sequences of rank 1 while the number of \(3\times 3\) matrices of rank 1 starting with column \((1 0 0)^T\) is 4. Table IV however remains valid since in that case \(r=k\).
In part B of Section IV \(\Psi\) is not defined, probably it is \(\Psi^1\). I do not understand how formula (24) is derived, and have some doubts on its correctness. The proofs of Lemma 13 and 14 are unclear. In Example 5 of Section E, \(x\to x^3\) is used as a permutation polynomial of the intermediate ring, but this is not stated. The sequence in Example 6 is shifted the same way as the sequence of table III.
Reviewer: H.J.Tiersma (Diemen)
MSC:
| 94A55 | Shift register sequences and sequences over finite alphabets in information and communication theory |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
