On the minimum distance of q-ary nonlinear codes - CanaDAM

37 Pages·2013·0.38 MB·English

Ontheminimumdistanceofq-ary nonlinearcodes
On the minimum distance of q-ary
nonlinear codes
Jaume Pujol, Merc`e Villanueva, Fanxuan Zeng
Combinatorics, CodingandSecurityGroup(CCSG)
Department ofInformationandCommunications Engineering (dEIC)
Autonomous UniversityofBarcelona (UAB)
June 10-13, CanaDAM 2013, St.John’s
Ontheminimumdistanceofq-ary nonlinearcodes
Outline
Introduction
Minimum weight and minimum distance
Improvements on minimum weight and minimum distance
Minimum weight and minimum distanceof q-arycodes
Conclusions and future research
Ontheminimumdistanceofq-ary nonlinearcodes
Introduction
Introduction
Hamming Weight: or simply weight, of a vector v =(v1,...,vn)
is the number of its entries which are nonzero.
Hamming Distance: or simply distance, between two vectors
v = (v1,...,vn) and w = (w1,...,wn) is the number of entries
where they differ.
An (n,M,d) code C is a subset of Fn, where F is a finite field of
q q q
q elements.
We start from Binary Codes, which are subsets of Fn.
2
Ontheminimumdistanceofq-ary nonlinearcodes
Introduction
Introduction
Given a code C, the problem of storing C in memory and finding
out the minimum (Hamming) weight w(C) and minimum
(Hamming) distance d(C) are well-known problems.
• If C is linear, then it can be compactly represented using a
generator matrix. Moreover, d(C) =w(C).
• If C is nonlinear, then a solution would be to know whether it
has another structure or not.
For example, there are binary codes which have a Z4-linear or
Z2Z4-linear structure and, therefore, they can also be compactly
represented using a quaternary generator matrix.
Ontheminimumdistanceofq-ary nonlinearcodes
Introduction
Introduction
Given a code C, the problem of storing C in memory and finding
out the minimum (Hamming) weight w(C) and minimum
(Hamming) distance d(C) are well-known problems.
• If C is linear, then it can be compactly represented using a
generator matrix. Moreover, d(C) =w(C).
• If C is nonlinear, then a solution would be to know whether it
has another structure or not.
For example, there are binary codes which have a Z4-linear or
Z2Z4-linear structure and, therefore, they can also be compactly
represented using a quaternary generator matrix.
Ontheminimumdistanceofq-ary nonlinearcodes
Introduction
Introduction
In general, binary codes without any of these structures can be
represented as the union of cosets of a binary linear subcode of C.
t
C = K +c ,
i
i[=0(cid:16) (cid:17)
where c0 =0, t+1 = M/2k, M = |C| and
K = {x ∈ Zn | x+C = C}.
2
The kernel K is the largest linear subcode for which it is true.
The coset leaders are given by the set of t vectors: {c1,...,ct}.
Ontheminimumdistanceofq-ary nonlinearcodes
Introduction
Introduction
Since K is linear, the binary code C can be represented by the
generator matrix of K and the set of coset leaders instead of a set
of all its codewords.
Then, the code C take up a memory space of order O(n(k+t)).
Example 1
Memory space of binary codes of length n, M = 219 codewords
and different dimension of kernel:
dimension of kernel k 0 1 ... 17 19
number coset leaders t 219 218−1 ... 3 0
memory space 219n 218n ... 20n 19n
Ontheminimumdistanceofq-ary nonlinearcodes
Minimumweightandminimumdistance
Minimum weight and distance
• For linear codes, the best algorithm to obtain the minimum
weight/distance based on the enumeration of some codewords is
the Brouwer-Zimmermann algorithm.
• For nonlinear codes, there is no other algorithms but the
enumeration of all codeswords, that is, the brute force algorithm.
Ontheminimumdistanceofq-ary nonlinearcodes
Minimumweightandminimumdistance
A short introduction to Brouwer-Zimmermann’s Algorithm
(cid:3)(cid:5)
(cid:2)
(cid:1)
(cid:3) (cid:1)
(cid:1)
(cid:1)(cid:2)(cid:2)
(cid:3)
(cid:3) (cid:1)
(cid:2) (cid:2)
(cid:1)(cid:1)(cid:1) (cid:4)(cid:5)(cid:6)(cid:5)(cid:7)(cid:8)(cid:7)(cid:9)(cid:10)(cid:11)(cid:5)(cid:12)(cid:13)(cid:14)(cid:9)(cid:15)(cid:16)(cid:9)(cid:11)(cid:6)(cid:8)(cid:7)(cid:11)(cid:3)(cid:17)(cid:14)(cid:11)(cid:18)(cid:9)(cid:19)(cid:11)(cid:20)(cid:14)(cid:15)(cid:3)(cid:21)
(cid:3) (cid:1)
(cid:2) (cid:2)
(cid:4)(cid:5)(cid:6)(cid:5)(cid:7)(cid:8)(cid:7)(cid:9)(cid:10)(cid:11)(cid:5)(cid:12)(cid:13)(cid:14)(cid:9)
(cid:1)(cid:1)(cid:1)
(cid:3) (cid:1)
(cid:4) (cid:4) (cid:22)(cid:15)(cid:10)(cid:11)(cid:3)(cid:9)(cid:23)(cid:15)(cid:8)(cid:6)(cid:18)(cid:9)(cid:15)(cid:16)(cid:9)(cid:8)(cid:6)(cid:11)(cid:6)(cid:8)(cid:7)(cid:11)(cid:3)(cid:17)(cid:14)(cid:11)(cid:18)(cid:9)(cid:19)(cid:11)(cid:20)(cid:14)(cid:15)(cid:3)(cid:21)
(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:13)(cid:24)(cid:3)(cid:25)(cid:26)(cid:27)
Ontheminimumdistanceofq-ary nonlinearcodes
Minimumweightandminimumdistance
Minimum weight and distance
Extend Coset: Given a binary code C and a vector v, denote
K = K ∪(K +v) as an extend coset of C. Since K is linear,
v
then K is also linear, and can be constructed as hBasis(K),vi.
v
t
Let C = K +c with t ≥ 2.
i=0 i
S (cid:0) (cid:1)
Proposition / Algorithm 1 (MinW)
The minimum weight of C is min({w(K ) | 1 ≤ i ≤ t}).
ci
Proposition / Algorithm 2 (MinD)
The minimum distance of C is
min({w(Kci) | 1 ≤ i≤ t}∪{w(Kci+cj) | 1 ≤ i < j ≤ t}).