Optimal Families of Perfect Polyphase Sequences From the Array Structure of Fermat-Quotient Sequences

Park, Ki-Hyeon; Song, Hong‐Yeop; Ким, Дае Сан; Golomb, Solomon W.

doi:10.1109/tit.2015.2511780

Cited by 26 publications

(12 citation statements)

References 42 publications

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“…Some pseudorandomness measures of the binary sequences derived from Fermat quotients are discussed in [5,7,12,16,18,27]. A collection of optimal families of perfect polyphase sequences using the Fermat quotient sequences is proposed in [28]. Note that (q p (u)) is a p 2 -periodic sequence over Z p , and we have the following result.…”

Section: 3mentioning

confidence: 97%

Hamming correlation of higher order

Su¹,

Winterhof²

2018

Advances in Mathematics of Communications

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We introduce a new measure of pseudorandomness, the (periodic) Hamming correlation of order which generalizes the Hamming autocorrelation (= 2). We analyze the relation between the Hamming correlation of order and the periodic analog of the correlation measure of order introduced by Mauduit and Sárközy. Roughly speaking, the correlation measure of order is a finer measure than the Hamming correlation of order. However, the latter can be much faster calculated and still detects some undesirable linear structures. We analyze examples of sequences with optimal Hamming correlation and show that they have large Hamming correlation of order for some very small > 2. Thus they have some undesirable linear structures, in particular in view of cryptographic applications such as secure communications.

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Section: 3mentioning

confidence: 97%

Hamming correlation of higher order

Su¹,

Winterhof²

2018

Advances in Mathematics of Communications

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“…Remark 2 Note that, for the proposed optimal PPS families of period p k , the size is always p − 1 even if the period p k of each sequence becomes longer. One interesting point is that, for the period p k , the optimal families constructed in [16], [17], [19], [20], [23], [24] are also of size p − 1. Until now, it is unclear whether p − 1 is the maximum achievable size or not.…”

Section: Corollarymentioning

confidence: 99%

“…After Sarwate's work, many constructions for PPS families, which achieve the Sarwate bound, have been proposed. These are based on the original Frank sequences [3], [17], [24] and the generalized Frank sequences [19], [23], the generalized chrip-like polyphase sequences [20]. Mow also considered optimal families of PPSs from his first unified construction [16] for some parameters.…”

mentioning

confidence: 99%

Optimal Families of Perfect Polyphase Sequences from Cubic Polynomials

Song

Song²

2018

IEICE Trans. Fundamentals

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For an odd prime p and a positive integer k ≥ 2, we propose and analyze construction of perfect p k -ary sequences of period p k based on cubic polynomials over the integers modulo p k . The constructed perfect polyphase sequences from cubic polynomials is a subclass of the perfect polyphase sequences from the Mow's unified construction. And then, we give a general approach for constructing optimal families of perfect polyphase sequences with some properties of perfect polyphase sequences and their optimal families. By using this, we construct new optimal families of p k -ary perfect polyphase sequences of period p k . The constructed optimal families of perfect polyphase sequences are of size p − 1. key words: perfect polyphase sequences, cubic polynomials, optimal families of perfect polyphase sequences 1. Introduction Various sequences have been widely used in modern digital communication systems [8], [10], [11] and radar systems [14]. Recent applications include such commercial mobile communication systems as CDMA, WCDMA, and 3GPP LTE [1] and global navigation satellite systems as GPS [2] and GALILEO [7]. These applications require sequences with good correlation, or the minimum possible correlation magnitude for all the non-trivial phase shifts. Thus, it is best that sequences for these applications have zero autocorrelation at all out-of-phases, and such sequences are called perfect sequences. A sequence is called a polyphase sequence if all the symbols are on the complex unit circle [8], [16], [18]. For many decades, perfect polyphase sequences (PPSs) have attracted engineers and researchers [5], [6], [9], [12], [15], [16], [18]-[20], [24]. For an integer N ≥ 2, Frank and Zadoff constructed N-ary sequences of period N 2 in 1962 [9]. There is another class of PPSs, called the Zadoff-Chu sequence, whose phase sequence are generated by a quadratic polynomials. This class was proposed by Chu in 1972 [5], and generalized by Popovic [20]. A generalized version of Chu's sequences was named generalized chirp-like sequences by Popovic. In [16], [18], Mow classified all the known PPSs into four classes: the generalized Frank

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“…Then the k-error linear complexity was determined for binary sequences derived from the polynomial quotient modulo a prime [5] or its power [22], respectively. In [23], a series of optimal families of perfect polyphase sequences were derived from the array structure of Fermat-quotient sequences.…”

Section: Introductionmentioning

confidence: 99%