We apply the Hadamard equivalence to all the binary matrices of the size mn and study various properties of this equivalence relation and its classes. We propose to use HR-minimal as a representative of each equivalence class, and count and/or estimate the number of HR-minimals of size mn. Some properties and constructions of HR-minimals are investigated. Especially, we figure that the weight on an HR-minimal's second row plays an important role, and introduce the concept of Quasi-Hadamard matrices (QH matrices). We show that the row vectors of mn QH matrices form a set of m binary vectors of length n whose maximum pairwise absolute correlation is minimized over all such sets. Some properties, existence, and constructions of Quasi-orthogonal sequences are also discussed. We also give a relation of these with cyclic difference sets. We report lots of exhaustive search results and open problems, one of which is equivalent to the Hadamard conjecture.

We show that the non-trivial correlation of two properly chosen column sequences of length q-1 from the array structure of two Sidelnikov sequences of periods qe-1 and qd-1, respectively, is upper-bounded by $(2d-1)sqrt{q} + 1$, if $2leq e < d < rac{1}{2}(sqrt{q}-rac{2}{sqrt{q}}+1)$. Based on this, we propose a construction by combining properly chosen columns from arrays of size $(q-1) imes rac{q^e-1}{q-1}$ with e=2,3,...,d. The combining process enlarge the family size while maintaining the upper-bound of maximum non-trivial correlation. We also propose an algorithm for generating the sequence family based on Chinese remainder theorem. The proposed algorithm is more efficient than brute force approach.

In this paper, we calculate autocorrelation of new generalized cyclotomic sequences of period pn for any n > 0, where p is an odd prime number.

We investigate binary sequence pairs with two-level correlation in terms of their corresponding cyclic difference pairs (CDPs). We define multipliers of a cyclic difference pair and present an existence theorem for multipliers, which could be applied to check the existence/nonexistence of certain hypothetical cyclic difference pairs. Then, we focus on the ideal case where all the out-of-phase correlation coefficients are zero. It is known that such an ideal binary sequence pair exists for length υ = 4u for every u ≥ 1. Using the techniques developed here on the theory of multipliers of a CDP and some exhaustive search, we are able to determine that, for lengths υ ≤ 30, (1) there does not exist "any other" ideal/ binary sequence pair and (2) every example in this range is equivalent to the one of length υ = 4u above. We conjecture that if there is a binary sequence pair with an ideal two-level correlation then its in-phase correlation must be 4. This implies so called the circulant Hadamard matrix conjecture.

In this paper, we investigate how to obtain binary locally repairable codes (LRCs) with good locality and availability from binary Simplex codes. We first propose a Combination code having the generator matrix with all the columns of positive weights less than or equal to a given value. Such a code can be also obtained by puncturing all the columns of weights larger than a given value from a binary Simplex Code. We call by block-puncturing such puncturing method. Furthermore, we suggest a heuristic puncturing method, called subblock-puncturing, that punctures a few more columns of the largest weight from the Combination code. We determine the minimum distance, locality, availability, joint information locality, joint information availability of Combination codes in closed-form. We also demonstrate the optimality of the proposed codes with certain choices of parameters in terms of some well-known bounds.

In this paper, we propose a framework to allocate code rates of component codes in a Plotkin-type unequal error protection (UEP) code. We derive an equivalent noise variance for each component code using structure of the Plotkin construction and Gaussian assumption. Comparing the equivalent noise variance and Shannon limit, we can find a combination of the code rates for the component codes. We investigate three types of code rate combinations and analyse their UEP performance. We also estimate a performance crossing signal to noise ratio (SNR) of the Plotkin-type UEP code. It indicates that which code has better performance for a given SNR. We confirm that the proposed framework is appropriate to obtain a desired UEP capability.

We define a new quaternary cyclotomic sequences of length 2p, where p is an odd prime. We compute the autocorrelation of these sequences. In terms of magnitude, these sequences have the autocorrelations with at most 4 values.

In this paper, we examine the locality property of the original Fractional Repetition (FR) codes and propose two constructions for FR codes with better locality. For this, we first derive the capacity of the FR codes with locality 2, that is the maximum size of the file that can be stored. Construction 1 generates an FR code with repetition degree 2 and locality 2. This code is optimal in the sense of achieving the capacity we derived. Construction 2 generates an FR code with repetition degree 3 and locality 2 based on 4-regular graphs with girth g. This code is also optimal in the same sense.

In this paper, we show that if the d-decimation of a (q-1)-ary Sidelnikov sequence of period q-1=pm-1 is the d-multiple of the same Sidelnikov sequence, then d must be a power of a prime p. Also, we calculate the crosscorrelation magnitude between some constant multiples of d- and d'-decimations of a Sidelnikov sequence of period q-1 to be upper bounded by (d+d'-1)√q+3.

Let PA(n, d) be a permutation array (PA) of order n and the minimum distance d. We propose a new construction of the permutation array PA(pm, pm-1k) for a given prime number p, a positive integer k < p and a positive integer m. The resulted array has (|PA(p,k)|p(m-1)(p-k))m rows. Compared to the other constructions, the new construction gives a permutation array of far bigger size with a large minimum distance, for example, when k ≥ 2p/3. Moreover the proposed construction provides an algorithm to find the i-th row of PA (pm, pm-1k) for a given index i very simply.

We propose three effective approximate belief propagation decoders for polar codes using Maclaurin's series, piecewise linear function, and stepwise linear function. The proposed decoders have the better performance than that of existing approximate belief propagation polar decoders, min-sum decoder and normalized min-sum decoder, and almost the same performance with that of original belief propagation decoder. Moreover, the proposed decoders achieve such performance without any optimization process according to the code parameters and channel condition unlike normalized min-sum decoder, offset min-sum decoder, and their variants.

For an odd prime p and a positive integer k ≥ 2, we propose and analyze construction of perfect pk-ary sequences of period pk based on cubic polynomials over the integers modulo pk. 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 pk-ary perfect polyphase sequences of period pk. The constructed optimal families of perfect polyphase sequences are of size p-1.