An ($N, n, s, t$) orthogonal array (OA) is an$N \times n$matrix over an$s$-ary alphabet, if for any choice of$t \leq n$columns, the resulting sub-array contains each$t$-tuple exactly$\lambda$times where$\lambda \times s^{t}=N$. The orthogonal array is an important object of study in many disciplines, namely the design of experiments, error-correcting codes, etc. The most important problem about orthogonal arrays is the following: What is the minimum number$N$of rows in an orthogonal array with$n$columns and strength$t$over an$s$-ary alphabet? Although there are several lower and upper bounds on the number$N$known in the literature, the gap between them is significantly large. In this work, we derive new inequalities and use them to improve the existing lower bound for some orthogonal arrays.

Golay complementary set (GCS) and its extension, the mutually orthogonal Golay complementary set (MOGCS), have important applications in modern wireless communication. Historically, GCS was used in multi-carrier systems such as orthogonal frequency division multiplexing (OFDM) to reduce the peak-to-mean envelope power ratio (PMEPR) of the unencoded signal via encoding. In contrast, the MOGCS has been used in multi-access systems to support a large pool of users in a cell in spread-spectrum communication. The recent decade has seen significant advancement in this area regarding the flexibility of the sequence parameters, such as set size, flock size, alphabet size, etc. Apart from that, many substitutes of GCSs and MOGCSs have also been introduced to serve extensive application requirements. Quite a few times, the constructions of these sequence sets are based on an existing MOGCS. Hence, MOGCS has theoretical importance in the sequence design area, apart from its applied significance. However, it is still an open problem to construct a complete MOGCS with respect to the theoretical set size bound, having flexible parameters. In this paper, we search for MOGCS with as much flexibility as possible in the parameter choices.

In many scenarios, there is a need to design error correcting codes that are required to correct erasures of certain types, unlike the random erasures observed at the output of a binary erasure channel. This has led us to study codes based on matrices.In this case, the code is described by a collection of matrices where the distance between two matrices is defined suitably so that certain type of erasure patterns can be corrected. For example, suppose a matrix is chosen from this collection and the entries in the first two rows and the first two columns of this matrix are erased. Can we still recover the original matrix? In this paper, we study such codes. The application of these codes to distributed storage is described.

Secret sharing scheme is a method in which the secret is divided among finitely many participants by a dealer such that only the legitimate set of participants can recover the secret. The collection of sets of legitimate participants is called the access structure of the secret sharing scheme. There are various ways of constructing secret sharing schemes. Determination of the access structure for a secret sharing scheme is an important problem. Most of the known secret sharing schemes are based on linear codes. A major drawback of secret sharing schemes based on linear codes is that these schemes are susceptible to Tompa- Woll attack. In this paper, we use nonlinear codes to construct secret sharing schemes. These secret sharing schemes perform better than the secret sharing schemes based on linear codes with respect to Tompa-Woll attack.

Rate-1/2 Complex orthogonal designs (CODs) is an important class of space-time block codes due to the fact that the decoding delay of these codes is substantially less than that of the maximal-rate CODs. Determining the least decoding delay of rate-1/2 CODs remains an important open problem. Nonetheless, a lower bound on the decoding delay of the rate-1/2 CODs with 2m columns is obtained under certain assumptions and codes have been constructed with the decoding delay meeting the lower bound for m = 1, 2 and 3 modulo 4. In this paper, it is shown that the same lower bound for the decoding delay holds even for more general class of rate-1/2 CODs. Furthermore, the construction of rate-1/2 CODs with decoding delay meeting the lower bound is provided for all m.

The maximal rate of a nonsquare complex orthogonal design forntransmit antennas is [1/2]+[1/(n)] ifnis even and [1/2]+[1/(n+1)] ifnis odd and the codes have been constructed for allnby Liang (2003) and Lu (2005) to achieve this rate. A lower bound on the decoding delay of maximal-rate complex orthogonal designs has been obtained by Adams (2007) and it is observed that Liang's construction achieves the bound on delay fornequal to 1 and 3 modulo 4 while Lu et al.'s construction achieves the bound forn=0 , 1, 3 mod 4. Forn=2 mod 4, Adams (2010) have shown that the minimal decoding delay is twice the lower bound, in which case, both Liang's and Lu et al.'s construction achieve the minimum decoding delay. For large value ofn, it is observed that the rate is close to half and the decoding delay is very large. A class of rate-[1/2] codes with low decoding delay for allnhas been constructed by Tarokh (1999). In this paper, another class of rate-[1/2] codes is constructed for allnin which case the decoding delay is half the decoding delay of the rate-[1/2] codes given by Tarokh This is achieved by giving first a general construction of square real orthogonal designs which includes as special cases the well-known constructions of Adams, Lax, and Phillips and the construction of Geramita and Pullman, and then making use of it to obtain the desired rate-[1/2] codes. For the case of nine transmit antennas, the proposed rate-[1/2] code is shown to be of minimal delay. The proposed construction results in designs with zero entries which may have high peak-to-average power ratio and it is shown that by appropriate postmultiplication, a design with no zero entry can be obtained with no change in the code parameters.

It is known that the Entropy Power Inequality (EPI) always holds if the random variables have density. Not much work has been done to identify discrete distributions for which the inequality holds with the differential entropy replaced by the discrete entropy. Harremoës and Vignat showed that it holds for the pair (B(m, p),B(n, p)), m, n ∈ ℕ, (where B(n, p) is a Binomial distribution with n trials each with success probability p) for p = 0.5. In this paper, we considerably expand the set of Binomial distributions for which the inequality holds and, in particular, identify n0(p) such that for all m, n ≥ n0(p), the EPI holds for (B(m, p),B(n, p)). We further show that the EPI holds for the discrete random variables that can be expressed as the sum of n independent and identically distributed (IID) discrete random variables for large n.

Space-time block codes based on orthogonal designs are useful from practical point of view as these codes admit symbol-by-symbol maximum-likelihood decoding and also achieve full diversity. Though the square orthogonal designs have been studied quite extensively, there is very little progress in the area of non-square orthogonal designs. Some of the results pertaining to rectangular orthogonal designs have been obtained through various combinatorial methods which are quite involved and complicated. In this paper, we study non-square orthogonal designs using set partition methods. Using this method, some of the problems in complex orthogonal designs have been addressed. Moreover, the proposed method provides a simple and an elegant approach for the study of rectangular designs.

Space-time block codes based on orthogonal designs are used for wireless communications with multiple transmit antennas which can achieve full transmit diversity and have low decoding complexity. However, the rate of the square real/complex orthogonal designs tends to zero with increase in number of antennas, while it is possible to have a rate-1 real orthogonal design (ROD) for any number of antennas. In case of complex orthogonal designs (CODs), rate-1 codes exist only for 1 and 2 antennas. In general, For n transmit antennas, the maximal rate of a COD is 1/2 + 1/n or 1/2 + 1/n+1 for n even or odd respectively. In this paper, we present a simple construction for maximal-rate CODs for any number of antennas from square CODs which resembles the construction of rate-1 RODs from square RODs. These designs are shown to be amenable for construction of a class of generalized CODs (called Coordinate-Interleaved Scaled CODs) with low peak-to-average power ratio (PAPR) having the same parameters as the maximal-rate codes. Simulation results indicate that these codes perform better than the existing maximal rate codes under peak power constraint while performing the same under average power constraint.

Space-Time Block Codes from square complex orthogonal designs (SCOD) have been extensively studied and most of the existing SCODs contain large number of zero. The zeros in the designs result in high peak-to-average power ratio (PAPR) and also impose a severe constraint on hardware implementation of the code when turning off some of the transmitting antennas whenever a zero is transmitted. Recently, rate 1/2 SCODs with no zero entry have been reported for 8 transmit antennas. In this paper, SCODs with no zero entry for 2atransmit antennas whenever a + 1 is a power of 2, are constructed which includes the 8 transmit antennas case as a special case. More generally, for arbitrary values of a, explicit construction of 2atimes 2arate a+1/2aSCODs with the ratio of number of zero entries to the total number of entries equal to 1 - aplusmn1/2a2[logs2(2a/a+1)]is reported, whereas for standard known constructions, the ratio is 1- a+1/2a. The codes presented do not result in increased signaling complexity. Simulation results show that the codes constructed in this paper outperform the codes using the standard construction under peak power constraint while performing the same under average power constraint.

Space-time block codes (STBCs) obtained from non-square complex orthogonal designs are bandwidth efficient compared to those from square real/complex orthogonal designs for colocated coherent MIMO systems and has other applications in (i) non-coherent MIMO systems with non-differential detection, (ii) Space-Time-Frequency codes for MIMO-OFDM systems and (iii) distributed space-time coding for relay channels. Liang (IEEE Trans. Inform. Theory, 2003) has constructed maximal rate non-square designs for any number of antennas, with rates given by [(a+1)/(2a)] when number of transmit antennas is 2a-1 or 2a. However, these designs have large delays. When large number of antennas are considered this rate is close to 1/2. Tarokh et al (IEEE Trans. Inform. Theory, 1999) have constructed rate 1/2 non-square CODs using the rate-1 real orthogonal designs for any number of antennas, where the decoding delay of these codes is less compared to the codes constructed by Liang for number of transmit antennas more than 5. In this paper, we construct a class of rate-1/2 codes for arbitrary number of antennas where the decoding delay is reduced by 50% when compared with the rate-1/2 codes given by Tarokh et al. It is also shown that even though scaling the variables helps to lower the delay it can not be used to increase the rate.

Space-Time Block Codes from square complex orthogonal designs (SCOD) have been extensively studied and most of the existing SCODs contain large number of zero resulting in high peak to average of power ratio (PAPR) and also impose a severe constraint on hardware implementation of the code when turning off some of the transmitting antennas whenever a zero is transmitted. Recently, maximal-rate SCODs with no zero entry have been reported for 8 transmit antennas, constructed using Amicable Orthogonal Designs. In this paper, maximal-rate SCODs with no zero entry for 2atransmit antennas whenever a+1 is a power of 2 are constructed which includes the 8 transmit antennas case as a special case. More generally, for arbitrary values of a, explicit construction of 2atimes 2amaximal-rate SCODs with the fraction of number of zero entries to the total number of entries equal to [1-(a-1)/(2a)] 2[log2(2a/a+1)]is reported. Moreover, it is shown that these codes are obtainable by premultiplying the well known SCODs with appropriate scaled unitary matrices consisting of nonzero entries that are plusmn1 only. A procedure to compute these scaled unitary matrices is also presented.

For ptimesn complex orthogonal designs in k variables, where p is the number of channels uses and n is the number of transmit antennas, the maximal rate k/p of the design is asymptotically half as n increases. But, for such maximal rate codes, the decoding delay p increases exponentially. To control the delay, if we put the restriction that p = n, i.e., consider only the square designs, then, the rate decreases exponentially as n increases. This necessitates the study of the maximal rate of the designs with restrictions of the form p = n+1, p = n+2, p = n+3 etc. In this paper, we study the maximal rate of complex orthogonal designs with the restrictions p = n+1 and p = n+2. We derive upper and lower bounds for the maximal rate for p = n+1 and p = n+2. Also for the case of p = n+1, we show that if the orthogonal design admit only the variables, their negatives and multiples of these by radic-1 and zeros as the entries of the matrix (other complex linear combinations are not allowed), then the maximal rate always equals the lower bound