اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدBounds for the $l_1$-distance of $q$-ary lattices obtained via Constructions D, D$^{}$ and $overline{D}$
175
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Lattices have been used in several problems in coding theory and cryptography. In this paper we approach $q$-ary lattices obtained via Constructions D, $D$ and $overline{D}$. It is shown connections between Constructions D and $D$. Bounds for the minimum $l_1$-distance of lattices $Lambda_{D}$, $Lambda_{D}$ and $Lambda_{overline{D}}$ and, under certain conditions, a generator matrix for $Lambda_{D}$ are presented. In addition, when the chain of codes used is closed under the zero-one addition, we derive explicit expressions for the minimum $l_1$-distances of the lattices $Lambda_{D}$ and $Lambda_{overline{D}}$ attached to the distances of the codes used in these constructions.
قيم البحث
اقرأ أيضاً
This paper is devoted to sequences and focuses on designing new two-dimensional (2-D) Z-complementary array pairs (ZCAPs) by exploring two promising approaches. A ZCAP is a pair of 2-D arrays, whose 2-D autocorrelation sum gives zero value at all tim
e shifts in a zone around the $(0,0)$ time shift, except the $(0,0)$ time shift. The first approach investigated in this paper uses a one-dimensional (1-D) Z-complementary pair (ZCP), which is an extension of the 1-D Golay complementary pair (GCP) where the autocorrelations of constituent sequences are complementary within a zero correlation zone (ZCZ). The second approach involves directly generalized Boolean functions (which are important components with many applications, particularly in (symmetric) cryptography). Along with this paper, new construction of 2-D ZCAPs is proposed based on 1-D ZCP, and direct construction of 2-D ZCAPs is also offered directly by 2-D generalized Boolean functions. Compared to existing constructions based on generalized Boolean functions, our proposed construction covers all of them. ZCZ sequences are a class of spreading sequences having ideal auto-correlation and cross-correlation in a zone around the origin. In recent years, they have been extensively studied due to their crucial applications, particularly in quasi-synchronous code division multiple access systems. Our proposed 2-D ZCAPs based on 2-D generalized Boolean functions have larger 2-D $mathrm{ZCZ}_{mathrm{ratio}}=frac{6}{7}$. Compared to the construction based on ZCPs, our proposed 2-D ZCAPs also have the largest 2-D $mathrm{ZCZ}_{mathrm{ratio}}$.
The $q$-ary block codes with two distances $d$ and $d+1$ are considered. Several constructions of such codes are given, as in the linear case all codes can be obtained by a simple modification of linear equidistant codes. Upper bounds for the maximum
cardinality of such codes is derived. Tables of lower and upper bounds for small $q$ and $n$ are presented.
We study the $k$-error linear complexity of subsequences of the $d$-ary Sidelnikov sequences over the prime field $mathbb{F}_{d}$. A general lower bound for the $k$-error linear complexity is given. For several special periods, we show that these sequences have large $k$-error linear complexity.
Efficient optimal prefix coding has long been accomplished via the Huffman algorithm. However, there is still room for improvement and exploration regarding variants of the Huffman problem. Length-limited Huffman coding, useful for many practical app
lications, is one such variant, in which codes are restricted to the set of codes in which none of the $n$ codewords is longer than a given length, $l_{max}$. Binary length-limited coding can be done in $O(n l_{max})$ time and O(n) space via the widely used Package-Merge algorithm. In this paper the Package-Merge approach is generalized without increasing complexity in order to introduce a minimum codeword length, $l_{min}$, to allow for objective functions other than the minimization of expected codeword length, and to be applicable to both binary and nonbinary codes; nonbinary codes were previously addressed using a slower dynamic programming approach. These extensions have various applications -- including faster decompression -- and can be used to solve the problem of finding an optimal code with limited fringe, that is, finding the best code among codes with a maximum difference between the longest and shortest codewords. The previously proposed method for solving this problem was nonpolynomial time, whereas solving this using the novel algorithm requires only $O(n (l_{max}- l_{min})^2)$ time and O(n) space.
Given a combinatorial $(d-1)$-sphere $S$, to construct a combinatorial $d$-sphere $S^{hspace{.2mm}prime}$ containing $S$, one usually needs some more vertices. Here we consider the question whether we can do one such construction without the help of
any additional vertices. We show that this question has affirmative answer when $S$ is a flag sphere, a stacked sphere or a join of spheres. We also consider the question whether we can construct an $n$-vertex combinatorial $d$-sphere containing a given $n$-vertex combinatorial $d$-ball.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
