24 research outputs found
A Class of Three-Weight Cyclic Codes
Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, a class of
three-weight cyclic codes over \gf(p) whose duals have two zeros is
presented, where is an odd prime. The weight distribution of this class of
cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a
subclass of the cyclic codes are also studied and proved to be optimal.Comment: 11 Page
A Family of Five-Weight Cyclic Codes and Their Weight Enumerators
Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, a family of -ary
cyclic codes whose duals have three zeros are proposed. The weight distribution
of this family of cyclic codes is determined. It turns out that the proposed
cyclic codes have five nonzero weights.Comment: 14 Page
Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension
The Walsh transform QˆQ^ of a quadratic function Q:Fpn→FpQ:Fpn→Fp satisfies |Qˆ(b)|∈{0,pn+s2}|Q^(b)|∈{0,pn+s2} for all b∈Fpnb∈Fpn , where 0≤s≤n−10≤s≤n−1 is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class C1C1 is defined for arbitrary n as C1={Q(x)=Trn(∑⌊(n−1)/2⌋i=1aix2i+1):ai∈F2}C1={Q(x)=Trn(∑i=1⌊(n−1)/2⌋aix2i+1):ai∈F2} , and the larger class C2C2 is defined for even n as C2={Q(x)=Trn(∑(n/2)−1i=1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2}C2={Q(x)=Trn(∑i=1(n/2)−1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2} . For an odd prime p, the subclass DD of all p-ary quadratic functions is defined as D={Q(x)=Trn(∑⌊n/2⌋i=0aixpi+1):ai∈Fp}D={Q(x)=Trn(∑i=0⌊n/2⌋aixpi+1):ai∈Fp} . We determine the generating function for the distribution of the parameter s for C1,C2C1,C2 and DD . As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case p>2p>2 , the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to C1C1 and C2C2 in terms of a generating function
Algebraic Codes For Error Correction In Digital Communication Systems
Access to the full-text thesis is no longer available at the author's request, due to 3rd party copyright restrictions. Access removed on 29.11.2016 by CS (TIS).Metadata merged with duplicate record (http://hdl.handle.net/10026.1/899) on 20.12.2016 by CS (TIS).C. Shannon presented theoretical conditions under which communication was possible
error-free in the presence of noise. Subsequently the notion of using error
correcting codes to mitigate the effects of noise in digital transmission was introduced
by R. Hamming. Algebraic codes, codes described using powerful tools from
algebra took to the fore early on in the search for good error correcting codes. Many
classes of algebraic codes now exist and are known to have the best properties of
any known classes of codes. An error correcting code can be described by three of its
most important properties length, dimension and minimum distance. Given codes
with the same length and dimension, one with the largest minimum distance will
provide better error correction. As a result the research focuses on finding improved
codes with better minimum distances than any known codes.
Algebraic geometry codes are obtained from curves. They are a culmination of years
of research into algebraic codes and generalise most known algebraic codes. Additionally
they have exceptional distance properties as their lengths become arbitrarily
large. Algebraic geometry codes are studied in great detail with special attention
given to their construction and decoding. The practical performance of these codes
is evaluated and compared with previously known codes in different communication
channels. Furthermore many new codes that have better minimum distance
to the best known codes with the same length and dimension are presented from
a generalised construction of algebraic geometry codes. Goppa codes are also an
important class of algebraic codes. A construction of binary extended Goppa codes
is generalised to codes with nonbinary alphabets and as a result many new codes
are found. This construction is shown as an efficient way to extend another well
known class of algebraic codes, BCH codes. A generic method of shortening codes
whilst increasing the minimum distance is generalised. An analysis of this method
reveals a close relationship with methods of extending codes. Some new codes from
Goppa codes are found by exploiting this relationship. Finally an extension method
for BCH codes is presented and this method is shown be as good as a well known
method of extension in certain cases
두 p진 데시메이션 수열 간의 상호상관도
학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2017. 2. 노종선.In this dissertation, the cross-correlation between two differently decimated sequences of a -ary m-sequence is considered. Two main contributions are as follows.
First, for an odd prime , , and a -ary m-sequence of period , the cross-correlation between two decimated sequences by and are investigated. Two cases of , with and with odd are considered. The value distribution of the cross-correlation function for each case is completely deterimined. Also, by using these decimated sequences, two new families of -ary sequences of period with good correlation property are constructed.
Second, an upper bound on the magnitude of the cross-correlation function between two decimated sequences of a -ary m-sequence is derived. The two decimation factors are and , where is an odd prime, , and . In fact, these two sequences corresponds to the sequences used for the construction of -ary Kasami sequences decimated by . The upper bound is given as .
Also, using this result, an upper bound of the cross-correlation magnitude between a -ary m-sequence and its decimated sequence with the decimation factor is derived.1 Introduction 1
1.1 Background 1
1.2 Overview of This Dissertation 7
2 Preliminaries 9
2.1 Finite Fields 9
2.2 Trace Functions and Sequences 11
2.3 Cross-Correlation Between Two Sequences 13
2.4 Characters and Weils Bound 15
2.5 Trace-Orthogonal Basis 16
2.6 Known Exponential Sums 17
2.7 Cross-Correlation of -ary Kasami Sequence Family 18
2.8 Previous Results on the Cross-Correlation for Decimations with 20
2.9 Cross-Correlation Between Two Decimated Sequences by and or 23
3 New -ary Sequence Families of Period with Good Correlation Property Using Two Decimated Sequences 26
3.1 Cross-Correlation for the Case of 27
3.2 Cross-Correlation for the Case of 37
3.3 Construction of New Sequence Families 43
4 Upper Bound on the Cross-Correlation Between Two Decimated -ary Sequences 52
4.1 Cross-Correlation Between and 53
4.2 Cross-Correlation Between and 66
5 Conclusions 69
Bibliography 72
Abstract (In Korean) 80Docto