On the nonexistence of Barker arrays and related matters

Al-Quaddoomi, Sabah; Scholtz, R.A.

doi:10.1109/18.42220

Cited by 24 publications

(38 citation statements)

References 10 publications

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“…In this section we use the key constructions, Theorem 3.2 for covering EBSs and Theorem 4.3 for BSs, to obtain difference sets with parameters from the Hadamard family (1). Although many of the results were previously known our intention is to show that the various construction methods in the literature can be concisely brought into the unifying framework of this paper.…”

Section: Application To Hadamard Difference Setsmentioning

confidence: 97%

“…The study of difference sets is also deeply connected with coding theory because the code, over a field F, of the symmetric design corresponding to a (v, k, *, n)-difference set may be considered as the right ideal generated by D in the group algebra FG [29,32]. Difference sets in abelian groups are the natural solution to many problems of signal design in digital communications, including synchronisation [25], radar [1], coded aperture imaging [23,52], and optical image alignment [41]. For a recent survey of difference sets see Jungnickel [29].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Unifying Construction for Difference Sets

Davis

Jedwab

1997

Journal of Combinatorial Theory, Series A

105

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We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, *, n)=(2 2d+4 (2 2d+2 &1)Â3, 2 2d+1 (2 2d+3 +1)Â3, 2 2d+1 (2 2d+1 +1)Â3, 2 4d+2 ) for d 0. The construction establishes that a McFarland difference set exists in an abelian group of order 2 2d+3 (2 2d+1 +1)Â3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of order p r . This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly for r>1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylow p-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups. Academic Press

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Section: Application To Hadamard Difference Setsmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

A Unifying Construction for Difference Sets

Davis

Jedwab

1997

Journal of Combinatorial Theory, Series A

105

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“…Therefore for all 0 _< u i < si, S Lemma 2.1 (i) was obtained for the respective cases r = 1 and r = 2 by Turyn and Storer [54] and by Calabro and Wolf [10]. From (2.4) we may deduce that for any sl × ... × Sr binary array A, The case r = 2 of this congruence was used as a starting point for investigations into twodimensional Barker arrays by Alquaddoomi and Scholtz [1], Jedwab [30], [32] and Jedwab et al [34].…”

Section: Fundamentalsmentioning

confidence: 99%

Generalized perfect arrays and menon difference sets

Jedwab

1992

Des Codes Crypt

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Given an s I × ... x s r integer-valued array A and a (0, 1) vector z = (zl, • -., Zr), form the array A' from A by recursively adjoining a negative copy of the current array for each dimension i where zi = 1. A is a generalized perfect array type z if all periodic autocorrelation coefficients of A' are zero, except for shifts (u 1 ..... Ur) where u i =-0 (mod si) for all i. The array is perfect ifz = (0, .... 0) and binary if the array elements are all +1. A nontrivial perfect binary array (PBA) is equivalent to a Menon difference set in an abelian group.Using only elementary techniques, we prove various construction theorems for generalized perfect arrays and establish conditions on their existence. We show that a generalized PBA whose type is not (0, ..., 0) is equivalent to a relative difference set in an abelian factor group. We recursively construct several infinite families of generalized PBAs, and deduce nonexistence results for generalized PBAs whose type is not (0, ..., 0) from well-known nonexistence results for PBAs. A central result is that a PBA with 22Y32u elements and no dimension divisible by 9 exists if and only if no dimension is divisible by 2 y+2. The results presented here include and enlarge the set of sizes of all previously known generalized PBAs.

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“…Alquaddoomi and Scholtz [1] defined an s × t Barker array to be an s × t binary array A for which |C A (u, v)| ≤ 1 for all (u, v) = (0, 0). This generalises the notion of a Barker sequence from one dimension (the case s = 1 or t = 1) to two dimensions; see [10] and [11] Following [1], define the following function for an s × t array A = (a ij ):…”

Section: Introductionmentioning

confidence: 99%

“…Alquaddoomi and Scholtz [1] established Lemma 1.3 for binary arrays, and then used it to prove Proposition 1.4 for Barker arrays. This generalised the approach taken by Tuyrn and Storer in their classical paper [15] on the one-dimensional (sequence) case.…”

Section: Introductionmentioning

confidence: 99%