2017 2nd International Conference on Computer Science and Technology (CST 2017) ISBN: 978-1-60595-461-5
Construction of Some Binary LCD Codes
Qiang FU
*,a, Rui-hu LI, Liang-dong LV and Luo-bin GUO
School of Science, Air Force Engineering University, Xi'an, Shaanxi 710051, P. R. China
*Corresponding author
Keywords: LCD, Binary optimal code, Minimum distance.
Abstract. A q-aryl linear code C is called a linear complementary dual (LCD) code if
it meets its dual trivially. Binary LCD codes play a significant role for their advantage of low complexity for implementations against side-channel attacks. In this paper, we construct some LCD codes over F2 by puncturing, shortening and extending. Then
using some cyclic codes, we propose several series of optimal LCD codes with d=4, 5,
6, respectively.
Introduction Let 2n
F be the n-dimensional row vector space over the binary fieldF2. A binary
linear [ , ]n k code C is a k-dimensional subspace ofF2n. The weight w x( ) of x∈C is
the number of its nonzero coordinates, if the minimum weight of nonzero codeword’s in C is d, then C is denoted asC=[ , , ]n k d . Two binary codes C and C′ are equivalent
if one can be obtained from the other by permuting the coordinates. The dual code C of a linear code C is defined as ⊥
2
{ n T for all }
x F x y xy
⊥ = ∈ ∣ ⋅ = =0 y∈
C C .
A code C is self-orthogonal if C C⊆ ⊥ and self-dual ifC C . A code C is said to be = ⊥
complementary dual if ⊥ ={ }0
C C .
Definition 1.1[1,10] An [ , ]n k code C is called optimal if it has the highest weight among all the [ , ]n k codes, such a code is denoted as C=[ , , ( , )]n k d n ko , where
( , ) {
o
d n k =max d∣there is an n k d code . And denote [ , , ]q } d n kl( , )=max d{∣there is an
[ , , ]n k d LCD code . If there is a q } C=[ , , ( , )]n k d n kl q LCD code, we call C an optimal LCD code. If there is a C=[ , , ( , ) 1]n k d n kl − qLCD code, C is called a near optimal
LCD code.
distance. Ref. [10] proposes some bounds for LCD codes and some optimal LCD codes for 17≤ ≤ ≤k n 30.
This paper studies the constructions of binary LCD codes. In Section II, we give some preliminaries and notations on LCD codes. Section III will construct LCD [ , ]n k
codes by traditional methods, such as puncturing, shortening and extending. In Section IV, using some cyclic codes, we derive some series of optimal LCD codes with d = 4,
5, 6, respectively. The last Section is the conclusion.
Preliminaries and Notations
Let 2 101 011 =
S , 2 2 2
3
3 1 3 T
=
S 0 S
S
0 1 and then we have 1 2 1k 1 2 1k
T
k k k
k+
− −
=
S 0 S
S
0 1 . Let
1≤ ≤ −s k 1. M sk( ) is the matrix obtained by deleting the first 2s−1 columns from
the generator matrix Sk of k-dimensional Simplex code, we call the code generated
by M sk( ) as MacDonald code M D( , )k s [2]. If s= −k 1,denote M kk( −1) by Mk ,
denote M D( , )k s by M D( )k .
In [3], an equivalent condition for an LCD code is given as follows:
Proposition 2.1.Let C be a linear code. Let G be a generator matrix of C and H
a parity-check matrix. Then the three following properties are equivalent: 1. C is LCD,
2. the matrix HH is invertible, T
3. the matrix GG is invertible. T For a cyclic LCD code, we have
Proposition 2.2 If g x( ) is the generator polynomial of a q-ary [ , ]n k cyclic code
C of block length n ,then C is an LCD code if and only if g x( ) is self-reciprocal, satisfying g x( )=xn* (1/ )g x and all the manic irreducible factors of g x( ) have the same multiplicity in g x( )and inxn−1.
Construction of LCD Codes by Traditional Methods
In this section, we construct optimal LCD [n,5] codes and present some LCD codes
with good parameters.
A Construction of [n,5] LCD codes
Theorem 3.1 If 5≤ ≤n 30 and n≠24,28 , there exists an optimal LCD
[ ,5, ( ,5)]n d nl code and a near optimal LCD [ ,5, ( ,5)]n d nl code.
Proof. Case 1: Construction of LCD [n,5] codes for 5≤ ≤n 21
Let 5,5 4 4,1 1,4 1
I
G =
0 1 , 2 2,3 5,6 3,3 3 S G I = 0
0 ,
2 2,3 2,1
5,7 2,3 2 2,1
1,3 1,3 1
S G S = 0 0 0 0 1 1
.Deleting the
columns in G5,5, G5,7 inM5, denote the left columns by G5,11 and G5,9. Extending G5,7 by (0,0,0,1,0)T, we obtained
5,8
Let G5,21=(M5| )I5 . Puncturing the code C generated by G5,21 at index sets {2},
{1,2}, {1,2,4},, {1, 2, 3, 14}, {1, 2, 3, 4, 14}, {1, 2, 3, 4, 13,17}, {1, 2, 3, 4, 13, 17, 19}, {1, …, 5, 10, 13,16}, {1,…, 6, 9, 10, 15} and {1,…,6,9,11,14,…,16}, optimal LCD [ ,5, ( ,5)]n d nl codes are achieved for n=10 or 12≤ ≤n 21.
Case 2: LCD [ ,5]n codes with 22≤ ≤n 30
If 23≤ ≤n 25 or 27≤ ≤n 29, let G5,n =(M5|G5, 16n− ), these matrices G5,n generate
optimal [n,5] LCD codes with n≠24,28 and near optimal [24,5,10] and [28,5,12]
LCD codes.
Let (0,0,0,0,1)T
e= , G5,17 =( |e M5), then G5,17generates an optimal [17,5,8] code
with (5,175,17) 1 T
rank G G = . For n=22, 26, 30, let 5, 5,17 4, 17
17 | n n
n
G
G G −
−
=
0 .These three matrices G5,n generate optimal [n,5] LCD codes.
Hence, the theorem holds.
Next, we present some LCD codes by method of puncturing, shortening and extending. Using these LCD codes, we can complete a table for LCD codes with 1≤ ≤ ≤k n 30 , for details, please see [10]. We give the LCD codes as follows:
k=6: [17, 6, 6]P, [18, 6, 7]P, [20, 6, 8]P, [22,6, 9]P, [23, 6, 10]P, [26, 6, 11]P,
[29, 6, 12]P, [30, 6, 13]P;
k=7: [17, 7, 6]P, [19, 7, 7]P, [21, 7, 8]P,[23, 7, 9]J, [26, 7, 10]E, [28, 7, 11] P; k=8: [17,8,6]P, [21,8,7] S, [22,8,8]P,[25,8,9]P, [26,8,10]P;
k=9: [17,9,5]P, [19,9,6]E, [22,9,7],[25,9,8] P, [26,9,9]P, [28,9,10]E;
k=10: [19,10,5]P, [20,10,6]M, [23,10,7]P, [24,10,8]P, [27,10,9] P, [28,10,10] M; k=11: [20,11,5]P, [23,11,6]S, [24,11,7]S, [27,11,8]P, [29,11,9]P;
k=12: [21,12,5]E, [22,12,6]S, [25,12,7]S,[26,12,8]P ; k=13:[23,13,5]P, [25,13,6]P, [26,13,7]P,[30,13,8]P;
k=14, 15: [24,14,5]S, [25,14,6]S, [28,14,7]E,[25,15,5]P, [28,15,6]P; k=16, 17: [26,16,5]P, [27,16,6]P,[27,17,5]P,[30,17,6]P;
k=18, 19, 20: [28,18,5]S, [29,18,6]S, [29,19,5]S,[30,20,5]S.
Remark 2: The superscript labels “P”, “E” and “S” in the code represent the code
is obtained by using the method puncturing, extending, shortening a given code and “M” stands for the code is from the Database in Magma [13].
Optimal LCD Codes from Cyclic Codes
In this section, we will construct some optimal LCD codes with distance d=4, 5, 6 using
some optimal cyclic codes.
A Construction of optimal LCD codes with d=4
To construct LCD codes with d=4, we first construct four cyclic codes with code length n=15, 31, 63, 127. Four kinds of optimal [n,n-5,4], [n,n-6,4], [n,n-7,4] and [n,n-8,4]
LCD codes can be derived from these cyclic codes. Let ( ) 1 2 4 5
g x = + + +x x x over F2[ ] / (x x15−1) be the generator polynomial of a
columns and i rows of G10,15 for i =4, 8, we obtain optimal [11,6,4] and [7,2,4] LCD
codes. And shortening the [15,10,4] code at position {1, 3, 6, 9, 13, 14}, an optimal [9,4,4] LCD code is obtained.
The polynomial ( ) 6 5 3 2 1
g x =x + + + + +x x x x over F2[ ] / (x x31−1) generates an
optimal cyclic [31,25,4] code, denote its generator matrix by G21,31 . Let i=5, 7, 9, 15,
17, 21, deleting the last i columns and i rows of G21,31, we achieve optimal [31-i,25-i,4]
LCD codes.
The polynomial ( ) 1 2 6 7
g x = + + +x x x over F2[ ] / (x x63−1) generates an optimal
cyclic [63,56,4] code. Let i=6,10, 16, 18, 20, 24, 26, 32, 36, 38, 40, 42, 46, deleting the
last i columns and i rows ofG56,63 , we achieve optimal [63-i,56-i,4] LCD codes.
The polynomial ( ) 8 7 2 1
g x = + + +x x x over F2[ ] / (x x127−1) generates an optimal
cyclic [127,119,4] code. Let i=7, 11, 13, 17, 19, 23, 27, 29, 31, 33, 37, 45, 47,
49,55, 63, 73, 77, 79, 83, 89, 95, 97, 99, 101, 103, 109, deleting the last i columns
and i rows of G119,127, we achieve optimal [127-i,119-i,4] LCD codes.
Summarizing the results above, we have
Theorem 4.1 There exist optimal [n,n-5,4] LCD codes for n=7, 9, 11, optimal [n,n-6,4] LCD codes for n=10, 14, 16, 22,24, 26, optimal [n,n-7,4] LCD codes for n=17, 21, 23, 25, 27,31, 37, 39, 43, 45, 47, 53, 57 and [n,n-8,4] LCD codes for n=18,24, 26, 28, 30, 32, 38, 44, 48, 50, 54, 64, 72, 78, 80,82, 90, 94, 96, 98, 100, 104, 108, 110, 114, 116, 120.
Construction of Optimal LCD codes with d=5
Consider the optimal LCD codes with d=5, we need cyclic codes with d=5. Like the
discussion above, we first construct some cyclic codes.
The polynomial ( ) 8 7 6 4 2 1
g x = + + + + + +x x x x x x over F2[ ] / (x x17−1) generates
an optimal cyclic [17,9,5] code. Let i=6, 5, 1, deleting the last i columns and i rows
ofG9,17 , we achieve optimal [17-i,9-i,5] LCD codes.
The polynomial g x( ) 1= + + + + + +x2 x3 x5 x7 x8 x10over 31 2[ ] / (x x −1)
F generates
an optimal cyclic [31,21,5] LCD code. Let i=3, 2, 1, deleting the last i columns and i
rows of G21,31, we achieve optimal [31-i,21-i,5] LCD codes.
The polynomial ( ) 12 10 8 5 4 3 1
g x =x +x + + + + +x x x x over F2[ ] / (x x63−1) gene-
rates an optimal cyclic [63,51,5] code. Let i=12, 13, 14, deleting the last i columns and i rows of G51,63 , we achieve optimal [63-i,51-i,5] LCD codes.
The polynomial ( ) 14 10 8 4 3 1
g x =x +x + + + + +x x x x over F2[ ] / (x x127−1) gene-
rates an optimal cyclic [127,113,5] code. Let i=14, 15, 16, 17, 18, 20, 21, 23, 24, 28, 29,
30, 32, 33, 35, 36, 42, 43, 44, deleting the last i columns and i rows of G113,127 , we
achieve optimal [127-i,113-i,5] LCD codes.
Summarizing the results above, we have
Construction of Optimal LCD codes with d=6
A similar method applied to cyclic code with d=6, we can obtain [n,n-9], [n,n-11],
[n,n-13] and [n,n-15] LCD codes with d=6. In this subsection, we begin with a cyclic
[17,8,6] code.
The polynomial ( ) 1 3 4 5 6 9
g x = + + + + +x x x x x over F x2[ ] / (x17−1) generates
an optimal cyclic [17,8,6] LCD code. Extending this cyclic code, we get an optimal [18,8,6] code. Both [17,8,6] and [18,8,6] code are optimal LCD codes according to [4]. Let i=2, 4, deleting the last i columns and i rows of G8,17 , we achieve
optimal[17-i,8-i,6] LCD codes.
The polynomial ( ) 1 3 4 5 6 7 8 10 11
g x = + + + + + + + +x x x x x x x x +x over F x2[ ] /
33
(x −1)generates an optimal cyclic [33,22,6] LCD code. It is an optimal LCD code according to [4]. Let i=8, 2, deleting the last i columns and i rows ofG22,33 , we achieve
optimal [33-i,22-i,6] LCD codes.
Shortening the [33, 22, 6] code at position {1, 2}, { 1, 2,3, 6 }, { 1, 2, 3, 4, 6, 7}, {1, 2,3,4,5,6,7,8,9,10,11,12,13,16,17,18} in G22,33, respectively, optimal [17,6,6], [27, 16,
6], [29, 18, 6] and [31, 20, 6] LCD codes are obtained.
In a similar way, we will use six cyclic codes with code length n=129 to construct
optimal LCD [s,s-15,6] codes for odd s.
[image:5.612.129.480.384.539.2]We give a table to show every optimal codes obtained from the six cyclic codes with different generator polynomials.
Table 1. Optimal LCD [s,s-15,6] codes from cyclic [129,114,6] codes.
generator vector 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 1
s 123, 113, 111, 109, 107, 101, 87, 81, 71, 65, 63, 61
generator vector 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1
s 127, 125, 115, 113, 111, 109, 103, 89, 87, 77,
71, 65, 63, 61, 59, 57 generator vector 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1
s 127, 125, 119, 117, 111, 105, 99, 93, 91, 89, 87, 69
generator vector 1 0 1 0 1 1 1 0 0 1 1 1 0 1 0 1
s 119, 117, 107, 101, 99, 97, 87, 81, 79, 73, 71, 69, 67
generator vector 1 1 1 1 0 1 0 0 0 0 1 0 1 1 1 1
s 127, 125, 123, 113, 107, 89, 87, 77, 75, 73, 67, 65, 59, 57
generator vector 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1
s 115, 105, 103, 97, 95, 93, 87, 81, 79, 77, 67, 65 According to Table 1 and the discussion above, we have
Theorem 4.3There exist optimal [ ,n n−9,6] LCD codes for n=13, 15, 17, optimal
[
n n, −11, 6]
LCD codes for n=17, 25, 27,29, 31, 33, optimal[
n n, −13,6]
LCD codes for n=35, 37, 39, 55, 57, 59, 61 and 63, optimal[
n n, −15,6]
LCD codes for odd n,57≤ ≤n 127 and s≠83,85,121.
Conclusion
In this paper, we have discussed the construction of binary LCD codes. Firstly, we have constructed some [ , ]n k LCD codes using traditional technique: puncturing, shortening
cyclic codes, we derived several series of optimal LCD codes with d=4, 5, 6,
respectively.
Acknowledgement
This work is supported by National Natural Science Foundation of China under Grant No.11471011, “973” program No.2013CB834204 and Natural Science Foundation of Shaanxi under Grant No. 2015JM1023.
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![Table 1. Optimal LCD [s,s-15,6] codes from cyclic [129,114,6] codes.](https://thumb-us.123doks.com/thumbv2/123dok_us/288832.1029697/5.612.129.480.384.539/table-optimal-lcd-s-codes-from-cyclic-codes.webp)