Abstract We investigate unbiased weighing matrices of weight 9 and provide a construction method using mutually suitable Latin squares. For $$n \le 16$$ n ≤ 16 , we determine the maximum size among sets of mutually unbiased weighing matrices of order n and weight 9. Notably, our findings reveal that 13 is the smallest order where such pairs exist, and 16 is the first order for which a maximum class of unbiased weighing matrices is found.

Abstract We introduce quaternary modified four $$\mu $$ μ -circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four $$\mu $$ μ -circulant Hermitian self-dual codes. We also construct quaternary modified four $$\mu $$ μ -circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual [56, 28, 16] codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) [56, 28] codes. In addition, these codes imply the existence of a quantum [[56, 0, 16]] code.

We give new characterizations of the largest minimum weights among LCD codes of large dimensions. Using the characterizations, we completely determine the largest minimum weights among binary LCD codes of length n and dimension$n-6$, ternary LCD codes of length n and dimension$n-5$and quaternary Hermitian LCD codes of length n and dimension$n-4$for arbitrary n. We also determine the largest minimum weights among binary LCD codes of length n and dimension$n-7$, ternary LCD codes of length n and dimension$n-6$and quaternary Hermitian LCD codes of length n and dimension$n-5$with only some exceptions n.

The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions on the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As a consequence, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, an entanglement-assisted quantum error-correcting [[21s + 5, 3, 16s + 3; 21s + 2]] code with maximal entanglement is constructed for the first time from a quaternary Hermitian linear complementary dual [26, 3, 19] code.

For some extremal (optimal) odd unimodular lattice L in dimensions 12, 16, 20, 28, 32, 36, 40, and 44, we determine all integers k such that L contains a k-frame. This result yields the existence of an extremal Type I Zk-code of lengths 12, 16, 20, 32, 36, 40, and 44, and a near-extremal Type I Zk-code of length 28 for positive integers k with only a few exceptions.

Battery performance of a smartphone is high priority for smartphone users. Wireless networking such as WEB browsing, SNS, contents download consumes one third of energy consumption of smartphone use case. As 3G-4G interface spends much power, it may recommend that using WLAN is more efficient for battery life. The various power-saving methods for WLAN have been discussed, but such solutions need modification of network infrastructure. Therefore, it is difficult to adequate current system. In this paper, we present a cross-layer power save method which can coexist with related works. The proposed method controls TCP-acknowledgement transmission depends on wireless channel quality and network quality. This method does not need any modification to current infrastructure, and can save battery power without Network performance degradation.

A classification of quaternary Hermitian self-dual codes of length 20 is given. Using this classification, a classification of extremal quaternary Hermitian self-dual codes of length 22 is also given.

In this paper, we construct new binary singly even self-dual codes with larger minimum weights than the previously known singly even self-dual codes for several lengths. Several known construction methods are used to construct the new self-dual codes.

In this correspondence, we construct new self-dual codes over ${\BBF}_4, {\BBF}_5, {\BBF}_7$ with larger minimum weights than the previously known self-dual codes, using construction methods based on circulant and negacirculant matrices. In particular, a self-dual $[36,18,13]$ code over ${\BBF}_7$ is constructed which has a larger minimum weight than the previously known $[36,18]$ codes.

In this correspondence, we give some restrictions on weight enumerators of singly even self-dual [n,n/2,d] codes whose shadows have minimum weight d/2. As a consequence, we determine the weight enumerators for which there is an extremal singly even self-dual [40,20,8] code and an optimal singly even self-dual [50,25,10] code

In this paper, we study optimal self-dual codes and type IV self-dual codes over the ring F/sub 2//spl times/F/sub 2/ of order 4. We give improved upper bounds on minimum Hamming and Lee weights for such codes. Using the bounds, we determine the highest minimum Hamming and Lee weights for such codes of lengths up to 30. We also construct optimal self-dual codes and type IV self-dual codes.

New extremal self-dual codes of length 62 are constructed with weight enumerators of three different types. Two of these types were not represented by any known code up till now. All these codes possess an automorphism of order 15. Some of them are used to construct extremal self-dual codes of length 60 by the method of subtracting. By additional subtracting, an extremal self-dual [58, 29, 10] code was obtained having a weight enumerator which does not correspond to any code known so far.

Previously, Type II codes over F/sub 4/ have been introduced as Euclidean self-dual codes with the property that all Lee weights are divisible by four. In this paper, a number of properties of Type II codes are presented. We construct several extremal Type II codes and a number of extremal Type I codes. It is also shown that there are seven Type II codes of length 12, up to permutation equivalence.

In an attempt to improve the performance under a frequency selective fading environment, we propose an orthogonal frequency division multiple access (OFDM) system in which MTCM (multiple trellis coded modulation) is applied to a set of multiple subcarriers. The code is designed to enlarge the summation of the Euclidean distance and the product distance. We also propose a scheme for assigning the symbols to subcarriers adaptively based on the channel state information (CSI) fed back from the reception receive end, and evaluate the performance applying these two codes and the proposed scheme under a multipath Rayleigh fading environment.

In this correspondence, we investigate the covering radius of codes over Z/sub 4/ for the Lee and Euclidean distances in relation with those of binary nonlinear codes and lattices obtained by the Gray map and Construction A/sub 4/, respectively. We give several upper and lower bounds on covering radii, including Z/sub 4/-analogs of the sphere-covering bound, the packing radius bound, the Delsarte bound, and the redundancy bound. We show that any Euclidean-optimal Type II code of length 24 has covering radius 8 with respect to the Euclidean distance. We determine the covering radius of the Klemm codes with respect to the Lee distance. We derive lower bounds on the covering radii of the Niemeier lattices.

We study self-dual codes over the ring Z/sub 2k/ of the integers modulo 2k with relationships to even unimodular lattices, modular forms, and invariant rings of finite groups. We introduce Type II codes over Z/sub 2k/ which are closely related to even unimodular lattices, as a remarkable class of self-dual codes and a generalization of binary Type II codes. A construction of even unimodular lattices is given using Type II codes. Several examples of Type II codes are given, in particular the first extremal Type II code over Z/sub 6/ of length 24 is constructed, which gives a new construction of the Leech lattice. The complete and symmetrized weight enumerators in genus g of codes over Z/sub 2k/ are introduced, and the MacWilliams identities for these weight enumerators are given. We investigate the groups which fix these weight enumerators of Type II codes over Z/sub 2k/ and we give the Molien series of the invariant rings of the groups for small cases. We show that modular forms are constructed from complete and symmetrized weight enumerators of Type II codes. Shadow codes over Z/sub 2k/ are also introduced.

We study Type IV self-dual codes over the commutative rings of order 4. Gleason-type theorems of Type IV codes and their shadow codes are investigated. A mass formula of Type IV codes over these rings are given. We give a classification of Type TV codes over Z/sub 4/ and F2+uF/sub 2/ for reasonable lengths. We also construct a number of optimal Type TV codes.

The alphabet F/sub 2/+uF/sub 2/ is viewed here as a quotient of the Gaussian integers by the ideal (2). Self-dual F/sub 2/+uF/sub 2/ codes with Lee weights a multiple of 4 are called Type II. They give even unimodular Gaussian lattices by Construction A, while Type I codes yield unimodular Gaussian lattices. Construction B makes it possible to realize the Leech lattice as a Gaussian lattice. There is a Gray map which maps Type II codes into Type II binary codes with a fixed point free involution in their automorphism group. Combinatorial constructions use weighing matrices and strongly regular graphs. Gleason-type theorems for the symmetrized weight enumerators of Type II codes are derived. All self-dual codes are classified for length up to 8. The shadow of the Type I codes yields bounds on the highest minimum Hamming and Lee weights.

We give two computational results on binary self-dual codes. A number of extremal singly-even self-dual [68, 34, 12] codes with weight enumerators not known to exist, are constructed. For k/spl les/10, the relationship between extremal doubly-even self-dual codes of length 24 k, and certain singly-even self-dual codes of length 24 k-2 is presented.

In this correspondence, we give a characterization of certain quasi-cyclic self-complementary codes with parameters [120,9,56] and [136,9,64], some quasi-cyclic self-complementary codes are also constructed with parameters [496,11,240] and [528,11,256]. These codes are optimal in the sense that they meet the Grey-Rankin bound, new quasi-symmetric SDP (symmetric difference property) designs are constructed from these codes.

Previously, Type IV self dual codes over the ring F/sub 2/+uF/sub 2/ of order 4 have been introduced. In this paper, we construct five optimal Type IV self-dual codes which are the first examples of such codes for that length. We also give several new optimal type IV self-dual codes, and Type IV self-dual codes which improve the upper bounds for minimum weights.

We construct an extremal singly-even self-dual [62, 31, 12] code. The existence of such a self-dual code was not previously known. Combined with known results, the above code also determines all lengths for which there are self-dual codes with minimum weight 12.

There are two distinct weight enumerators for extremal singly-even self-dual [36, 18, 8] codes. For each weight enumerator, only one extremal self-dual code is previously known. In this correspondence, we construct several new extremal singly-even self dual [36, 18, 8] codes for each weight enumerator. We also construct a number of new extremal singly-even self-dual [38, 19, 8] codes.

Recently there has been tremendous interest in self-dual codes over finite rings, specifically the rings Z/sub 2k/. In this paper, we investigate double circulant self-dual codes over Z/sub 2k/ for small k and short lengths. In particular, we give a classification of the extremal double circulant codes. Using invariant theory, a basis for the space of invariants to which the Hamming weight enumerators belong is given for self-dual codes over Z/sub 6/, Z/sub 8/, Z/sub 10/, and Z/sub 12/.

In this correspondence, we investigate binary extremal self-dual codes. Numerous extremal self-dual codes and interesting self-dual codes with minimum weight d=14 and 16 are constructed. In particular, the first extremal Type I [86,43,16] code and new extremal self-dual codes with weight enumerators which were not previously known to exist for lengths 40,50,52 and 54 are constructed. We also determine the possible weight enumerators for extremal Type I codes of lengths 66-100.

Conway and Sloane (1990) have listed the possible weight enumerators for extremal self-dual codes up to length 72. In this correspondence, we construct extremal singly-even self-dual [60,30,12] codes whose weight enumerator does not appear in this list. In addition, we present the possible weight enumerators for extremal self-dual codes of length 60.