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

On the minimum distance of q-ary nonlinear codes

On the minimum distance of q-arynonlinear codes

Jaume Pujol, Merce Villanueva, Fanxuan Zeng

Combinatorics, Coding and Security Group (CCSG)Department of Information and Communications Engineering (dEIC)

Autonomous University of Barcelona (UAB)

June 10-13, CanaDAM 2013, St.John’s


Outline

Introduction

Minimum weight and minimum distance

Improvements on minimum weight and minimum distance

Minimum weight and minimum distance of q-ary codes

Conclusions and future research


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 vectorsv = (v1, ..., vn) and w = (w1, ..., wn) is the number of entrieswhere they differ.

An (n,M, d)q code C is a subset of Fnq , where Fq is a finite field of

q elements.We start from Binary Codes, which are subsets of Fn

2 .


Introduction

Introduction

Given a code C, the problem of storing C in memory and findingout 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 agenerator matrix. Moreover, d(C) = w(C).

• If C is nonlinear, then a solution would be to know whether ithas another structure or not.

For example, there are binary codes which have a Z4-linear orZ2Z4-linear structure and, therefore, they can also be compactlyrepresented using a quaternary generator matrix.


Introduction

Introduction

Given a code C, the problem of storing C in memory and findingout 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 agenerator matrix. Moreover, d(C) = w(C).

• If C is nonlinear, then a solution would be to know whether ithas another structure or not.

For example, there are binary codes which have a Z4-linear orZ2Z4-linear structure and, therefore, they can also be compactlyrepresented using a quaternary generator matrix.


Introduction

Introduction

In general, binary codes without any of these structures can berepresented as the union of cosets of a binary linear subcode of C.

C =

t⋃

i=0

(

K + ci

)

,

where c0 = 0, t+ 1 = M/2k, M = |C| and

K = {x ∈ Zn2 | x+ C = C}.

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}.


Introduction

Introduction

Since K is linear, the binary code C can be represented by thegenerator matrix of K and the set of coset leaders instead of a setof 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 codewordsand 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



Minimum weight and distance

• For linear codes, the best algorithm to obtain the minimumweight/distance based on the enumeration of some codewords isthe Brouwer-Zimmermann algorithm.

• For nonlinear codes, there is no other algorithms but theenumeration of all codeswords, that is, the brute force algorithm.



A short introduction to Brouwer-Zimmermann’s Algorithm

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Minimum weight and distance

Extend Coset: Given a binary code C and a vector v, denoteKv = K ∪ (K + v) as an extend coset of C. Since K is linear,then Kv is also linear, and can be constructed as 〈Basis(K), v〉.

Let C =⋃t

i=0

(

K + ci)

with t ≥ 2.

Proposition / Algorithm 1 (MinW)

The minimum weight of C is min({w(Kci) | 1 ≤ i ≤ t}).

Proposition / Algorithm 2 (MinD)

The minimum distance of C ismin({w(Kci) | 1 ≤ i ≤ t} ∪ {w(Kci+cj) | 1 ≤ i < j ≤ t}).



Explanation of Algorithm 2(MinD)

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Example 2

Let K be the binary linear code of length n = 31, dimensionk = 5, and d(K) = 16, constructed as the dual of the binaryHamming code of length n = 31. Let C =

⋃

3

i=0

(

K + ci)

, wherec0 = 0, and the coset leaders are:

c1 = (0010001110011010011110001011110)

c2 = (0101101010111100101110100111101)

c3 = (0000011100011101101000111101011)

It is easy to check that the minimum weight of C is w(C) = 10and its minimum distance d(C) = 8.

minimum weight minimum distance

Brute Force 0.00018s 0.00840s

Algorithm 1 / 2 0.00060s 0.00126s

Note that sometimes a brute force calculation can be a faster way



Example 2

Let K be the binary linear code of length n = 31, dimensionk = 5, and d(K) = 16, constructed as the dual of the binaryHamming code of length n = 31. Let C =

⋃

3

i=0

(

K + ci)

, wherec0 = 0, and the coset leaders are:

c1 = (0010001110011010011110001011110)

c2 = (0101101010111100101110100111101)

c3 = (0000011100011101101000111101011)

It is easy to check that the minimum weight of C is w(C) = 10and its minimum distance d(C) = 8.

minimum weight minimum distance

Brute Force 0.00018s 0.00840s

Algorithm 1 / 2 0.00060s 0.00126s

Note that sometimes a brute force calculation can be a faster way



Time of computing w(C) using Algorithm 1(MinW) comparedwith brute force, for binary nonlinear codes of length n = 100, sizeM = 215 · 31, and kernel of dimension k ∈ {7, . . . , 15}.

7 8 9 10 11 12 13 14 150

1

2

3

4

5

6

7

Dimension of kernel

Tim

e: s

Algorithm MinWBrute force



Time of computing d(C) using Algorithm 2(MinD) compared withbrute force, for binary nonlinear codes of length n = 100, sizeM = 27 · 31, and kernel of dimension k ∈ {3, . . . , 7}.

3 4 5 6 70

20

40

60

80

100

120

140

Dimension of kernel

Tim

e: s

Algorithm MinDBrute force



Work factor

All these algorithms are based on the enumeration of codewords,adding together codewords and deciding the minimum weight ofcodewords.

The performance measurement of these computations is referred aswork. An estimate of the total work an algorithm requires isreferred as work factor.



Work factor of binary linear codes

For a binary linear code K of length n and dimension k, theminimum weight and minimum distance are the same, the workfactor of computing them using Zimmermann algorithm is:

WFZim = (n− k)⌈n/k⌉

r∑

r=1

(

k

r

)

where r is the smallest natural number such that

⌊n/k⌋(r + 1) +max(0, r + 1− (k − n mod k)) ≥ w(C).



Work factor of minimum weight

For a binary nonlinear code of length n, dimension of kernel k andnumber of coset leaders t, we can estimate the work factor:

Brute Force Algorithm: Enumerating every codeword of the codeand examining their weights, the work factor is:

n ·M = n · (t+ 1) · 2k

Algorithm MinW: The work factor is:

t∑

i=1

(

(n− (k + 1))⌈n/(k + 1)⌉

ri∑

r=1

(

k + 1

r

)

)

where ri is the smallest natural number such that

⌊n/(k + 1)⌋(ri+1)+max(0, ri+1−(k+1−n mod k+1)) ≥ w(Kci).



Work factor of minimum distance

Brute Force Algorithm: Enumerating every pair of codewords ofthe code, and exam the distance between them, the work factor is:

n ·

(

M

2

)

= n ·

(

2k(t+ 1)

2

)

Algorithm MinD: The work factor is:

t−1∑

i=0

(

t∑

j=i+1

(

(n− (k + 1))⌈n/(k + 1)⌉

ri,j∑

r=1

(

k + 1

r

)

))

where ri,j is the smallest natural number such that

⌊n/(k + 1)⌋(ri,j+1)+max(0, ri,j+1−(k+1−n mod k+1)) ≥ w(Kci+cj).



Work factor upper bounds

The minimum weight of an extend coset Kv = K ∪ (K + v) relieson the minimum weight of itself, so it is hard to estimate the workfactor.

However, note that w(Kv) ≤ w(K), where w(K) is the minimumweight of the kernel. Therefore, we can obtain an upper bound byreplacing w(Kv) with w(K).



Work factor upper bounds

Minimum Weight Upper Bound:

t ·(

(n− (k + 1))⌈n/(k + 1)⌉r

∑

r=1

(

k + 1

r

)

)

Minimum Distance Upper Bound:

(

t+ 1

2

)

·(

(n− (k + 1)) · ⌈n/(k + 1)⌉

r∑

r=1

(

k + 1

r

)

)


⌊n/(k + 1)⌋(r+1)+max(0, r+1−(k+1−n mod k+1)) ≥ w(K).



Work factor and work factor upper bound of minimum weight forbinary codes of length n = 100, size M = 217 · 31, and dimensionof kernel k ∈ {7, . . . , 17}.

7 8 9 10 11 12 13 14 15 16 170

0.5

1

1.5

2

2.5x 10

9

Dimension of kernel

Wor

k F

acto

r

Upper bound of Algorighm MinWAlgorithm MinW



Work factor and work factor upper bound of minimum distance forbinary codes of length n = 100, size M = 27 · 31, and dimension ofkernel k ∈ {3, . . . , 7}.

3 4 5 6 70

2

4

6

8

10

12x 10

8

Dimension of kernel

Wor

k F

acot

r

Upper bound of Algorithm MinDAlgorithm MinD



Improvement on Algorithm MinW and MinD

According to Algorithm MinW and MinD, we can obtain theminimum weight and distance of a binary code with t coset leadersby getting the minimum weight of several extend cosets.

Therefore, the codewords in the kernel are checked several times:• t times for minimum weight in Algorithm MinW.•(

t+1

2

)

times for minimum distance in Algorithm MinD.



An improved algorithm is brought to avoid the repetition.

Algorithm 3 (IMinD)

1. Add the all-zero codeword into the set of coset leaders.2. Add a coset leader to the basis of the kernel to form a binarylinear code of rank k + 1(for the added all-zero coset leader, therank is still k).3. Use Gaussian elimination to make the coordinates ofinformation set on the coset leader be 0. Then delete thesecoordinates in all rows.4. During every stage of enumerating r rows in Zimmermann’salgorithm, select r rows from the basis of the kernel and alwaysselect the coset leader row.



Explanation of Algorithm 3(IMinD)

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Work factor and upper bounds of Algorithm IMinD

Minimum Distance:

t∑

i=0

(

t∑

j=i+1

(

(n− k)⌈n/k⌉

ri,j∑

r=1

(

k

r

)

))


⌊n/k⌋(ri,j + 1) +max(0, ri,j + 1− (k − n mod k)) ≥ w(Kci+cj).

Upper Bounds:

(

(

t+ 1

2

)

+ 1) · (n− k) · ⌈n/k⌉r

∑

r=1

(

k

r

)

.


⌊n/k⌋(r + 1) +max(0, r + 1− (k − n mod k)) ≥ w(K).



Work factor of minimum distance using Algorithm 2(MinW) and 3on codes of length n = 100, kernel k ∈ {7, . . . , 15}, sizeM = 217 · 31.

3 4 5 6 70

2

4

6

8

10

12

14x 10

8

Dimension of kernel

wor

k fa

ctor

Upper bound of Algorithm MinDAlgorithm MinDUpper bound of Algorithm IMinDAlgorithm IMinDBrute force



Minimum weight/distance of q-ary codes

Problem: If K is a linear code,Binary Codes: K ∪ (K + c) is a linear code which can beobtained by adding c in the generator matrix.q-ary Codes: K ∪ (K + c) is not necessarily a linear code, and itcan not be obtained by adding c in the generator matrix.

Lemma 3

For a [n, k, d]q code K and a vector v ∈ Fnq , the code

Kv = 〈Basis(K), v〉 is a linear code, and the minimum weightw(Kv) = w(K(v)), where K(v) = K ∪ (K + v).



Minimum weight/distance of q-ary codes

Let C =⋃t

i=0

(

K + ci)

be an (n,M, d)q code given by kernel Kand a list of coset leaders {c1, c2, ..., ct}.

Proposition / Algorithm 4 (MinWq)

The minimum weight of C is min({w(Kci) | 1 ≤ i ≤ t}).

Proposition / Algorithm 5 (MinDq)

The minimum distance of C ismin({w(Kci) | 1 ≤ i ≤ t} ∪ {w(Kci+cj) | 1 ≤ i < j ≤ t}).



Improved minimum weight/distance of q-ary codes

Let C =⋃t

i=0

(

K + ci)

be an (n,M, d)q code given by kernel Kand a list of coset leaders {c1, c2, ..., ct}.

Proposition / Algorithm 6 (IMinWq)

The minimum weight of C is min({w(K + ci) | 1 ≤ i ≤ t}).

Proposition / Algorithm 7 (IMinDq)

The minimum distance of C ismin({w(K + ci) | 1 ≤ i ≤ t} ∪ {w(K + ci + cj) | 1 ≤ i < j ≤ t}).

Note that for the q-ary codes, since the coset K + ci contains only1/q codewords as Kv, this improvement save much time.



Minimum distance work factor of Algorithm IMinDq

Work Factor:

t∑

i=0

(

t∑

j=i+1

(

(n− k)⌈n/k⌉

ri,j∑

r=1

(

k

r

)

(q − 1)r−1))


⌊n/k⌋(ri,j + 1) +max(0, ri,j + 1− (k − n mod k)) ≥ w(Kci+cj).

Upper Bound:

(

(

t+ 1

2

)

+ 1) · (n− k) · ⌈n/k⌉r

∑

r=1

(

k

r

)

(q − 1)r−1.


⌊n/k⌋(r + 1) +max(0, r + 1− (k − n mod k)) ≥ w(K).



Upper bounds of work factor of minimum distance using AlgorithmIMinDq and brute force on codes over F3 of length n = 100,dimension of kernel k ∈ {3, . . . , 9}, size M = 39 · 31, andw(K) = 35.

2 3 4 5 6 70

0.5

1

1.5

2

2.5

3x 10

11

Dimension of kernel

Wor

k fa

ctor

Upper bound of Algorithm IMinDqBrute force



Conclusions

• We applied the structure of the kernel and a set of coset leadersto represent a code C to store it more compactly.

• Based on this structure, we developed several algorithms todecide the minimum weight and minimum distance of a code C.

• We evaluated the performance of the algorithms in theory andpartly in practical test.



Future research

• Apply the algorithms, especially for the general q-ary codes.

• Try to find a point based on the parameters from where we candecide to use the improved algorithms.

• Use the minimum weight to decode nonlinear binary codes.

• Use these algorithms to look for better nonlinear codes with bigminimum distance.


Bibliography

H. Bauer, B. Ganter and F. Hergert, ”Algebraic techniques for nonlinearcodes,” Combinatorica, vol. 3, pp. 21-33, 1983.

J. J. Cannon and W. Bosma (Eds.), Handbook of Magma Functions,Edition 2.13, 4350 pages, 2006.

M. Grassl, Searching for linear codes with large minimum distance, in: W.Bosma and J. Cannon (Eds.) Discovering Mathematics with Magma,Springer, 2006.

O. Heden, “On perfect p-ary codes of length p+ 1,” Designs, Codes and

Cryptography, vol. 46, no. 1, pp. 45-56, 2008.

G. White, “Enumeration-based Algorithms in Coding Theory,” PhDThesis, University of Sydney, 2006.

K.-H. Zimmermann, “Integral Hecke Modules, Integral GeneralizedReed-Muller Codes, and Linear Codes,” Tech. Rep. 3-96, TechnischeUniversitat Hamburg-Harburg, 1996.

A. Betten, H. Fripertinger, A. Kerber, A. Wassermann, and K.-H.

Zimmermann, Codierungstheorie: Konstruktionen und Auwendungen

linearer Codes, Berlin: Springer, 1998.


Bibliography

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