Perfect Binary Array Research Articles

Superimposed Perfect Binary Array-Aided Channel Estimation for OTFS Systems.

Orthogonal time-frequency space (OTFS) modulation outperforms orthogonal frequency-division multiplexing in high-mobility scenarios through better channel estimation. Current superimposed pilot (SP)-based channel estimation improves the spectral efficiency (SE) when compared to that of the traditional embedded pilot (EP) method. However, it requires an additional non-superimposed EP delay-Doppler frame to estimate the delay-Doppler taps for the following SP-aided frames. To handle this problem, we propose a channel estimation method with high SE, which superimposes the perfect binary array (PBA) on data symbols as the pilot. Utilizing the perfect autocorrelation of PBA, channel estimation is performed based on a linear search to find the correlation peaks, which include both delay-Doppler tap information and complex channel gain in the same superimposed PBA frame. Furthermore, the optimal power ratio of the PBA is then derived by maximizing the signal-to-interference-plus-noise ratio (SINR) to optimize the SE of the proposed system. The simulation results demonstrate that the proposed method can achieve a similar channel estimation performance to the existing EP method while significantly improving the SE.

Constructions of binary array set with zero-correlation zone

Based on the perfect binary array and the orthogonal matrix, a new method of constructing zero-correlation zone binary array set is proposed, yielding binary array sets with member sizes almost achieve the theoretical upper bound. A recursive construction of zero-correlation zone binary array set is also proposed and the obtained array sets have even larger sizes as well as additional good correlation properties.

Family of two-dimensional correcting codes on a basis of perfect binary array

We consider a family of two-dimensional correcting E(N)-codes on a basis of perfect binary arrays H(N), and we show correcting possibilities of E(N)-codes in comparison with corresponding BCH-codes with maximal length are essentially better with regard to correction of package (correlated) errors, at that, uncorrelated errors are corrected identically.

Two-dimensional perfect quaternary arrays

We study two-dimensional (2-D) arrays of fourth roots of unity which have all out-of-phase periodic autocorrelations equal to zero. Generalizing the concept of a perfect binary array, we call these arrays perfect quaternary arrays. We establish connections with combinatorial design theory and exhibit large families of such arrays. Increasing the alphabet from size two to size four greatly increases the flexibility one has in choosing the dimensions for 2-D arrays with perfect periodic autocorrelation. For example, we show that the number of entries in a 2-D perfect quaternary array may be divisible by any Mersenne prime and indeed by many other primes, whereas 2-D perfect binary arrays are only known to exist with size equal to a power of two times a power of three.

Nested Hadamard difference sets

A Hadamard difference set (HDS) has the parameters (4 N 2, 2 N 2 − N, N 2 − N). In the abelian case it is equivalent to a perfect binary array, which is a multidimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. We show that if a group of the form H × Z 2 p r contains a ( hp 2 r , √ hp r (2√ hp r − 1), √ hp r (√ hp r − 1)) HDS (HDS), p a prime not dividing | H| = h and p j ≡ −1 (mod exp( H)) for some j, then H × Z 2 p t has a ( hp 2 t , √ hp t (2√ hp t − 1), √ hp t (√ hp t − 1)) HDS for every 0⩽ t⩽ r. Thus, if these families do not exist, we simply need to show that H × Z 2 p does not support a HDS. We give two examples of families that are ruled out by this procedure.

A nonexistence result for abelian menon difference sets using perfect binary arrays

A Menon difference set has the parameters (4N2,2N2-N, N2-N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group\(H \times K \times Z_{p^\alpha }\) contains a Menon difference set, wherep is an odd prime, |K|=pα, andpj≡−1 (mod exp (H)) for somej. Using the viewpoint of perfect binary arrays we prove thatK must be cyclic. A corollary is that there exists a Menon difference set in the abelian group\(H \times K \times Z_{3^\alpha }\), where exp(H)=2 or 4 and |K|=3α, if and only ifK is cyclic.

Perfect binary arrays and difference sets

A perfect binary array is an r-dimensional array with elements ± 1 such that all out-of-phase periodic autocorrelation coefficients are zero. Such an array is equivalent to a Menon difference set in an abelian group. We give recursive constructions for four infinite families of two-dimensional perfect binary arrays, using only elementary methods. Brief outlines of the proofs were previously given by three of the authors. Although perfect binary arrays of the same sizes as two of the families were constructed earlier by Davis, the sizes of the other two families are new.

Comment: ‘Nonexistence of certain perfect binary arrays’ and ‘Nonexistence of perfect binary arrays’

For original papers see ibid., vol.29, p.99-101 (1993) and ibid., vol.27, p.1252-1254 (1991). The commenter suggests that the nonexistence of the 4*3*3*9 perfect binary array (PBA) can be easily proved, without the need for a lengthy computer search. He points out that the nonexistence of the 9*36 PBA and 18*18 PBA can also be proved. In reply, the original authors suggest that the commenter's analysis is based on an incorrect lemma.

Non-existence of certain perfect binary arrays

A perfect binary array (PBA) is an r-dimensional matrix with elements ± 1 such that all out-of-phase periodic autocorrelation coefficients are zero. The two smallest sizes for which the existence of a PBA is undecided, 2 × 2 × 3 × 3 × 9 and 4 × 3 × 3 × 9, are ruled out using computer search and a combinatorial argument.

Generalized perfect arrays and menon difference sets

Given an s 1 × ... × s rinteger-valued array A and a (0, 1) vector z = (z 1, ..., z r), form the array A′ from A by recursively adjoining a negative copy of the current array for each dimension i where z i = 1. A is a generalized perfect array type z if all periodic autocorrelation coefficients of A′ are zero, except for shifts (u 1, ..., u r) where u i, - 0 (mod s i) for all i. The array is perfect if z = (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 22y 32u 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.

Existence of one-dimensional perfect binary arrays

The close relationship between perfect binary arrays and higher-dimensional Hadamard matrices is determined in the letter. With this relationship we have proved that there exists no one-dimensional perfect binary array with energy greater than four. We conjecture that this relationship may be helpful for the study of other higher-dimensional perfect binary arrays.