Let da(n,5) and dl(n,5) be the minimum weights of optimal binary [n,5] linear codes and linear complementary dual (LCD) codes, respectively. This article aims to investigate dl(n,5) of some families of binary [n,5] LCD codes when n=31s+t≥14 with s integer and t∈{2,8,10,12,14,16,18}. By determining the defining vectors of optimal linear codes and discussing their reduced codes, we classify optimal linear codes and calculate their hull dimensions. Thus, the non-existence of these classes of binary [n,5,da(n,5)] LCD codes is verified, and we further derive that dl(n,5)=da(n,5)−1 for t≠16 and dl(n,5)=16s+6=da(n,5)−2 for t=16. Combining them with known results on optimal LCD codes, dl(n,5) of all [n,5] LCD codes are completely determined.
Citation: Yang Liu, Ruihu Li, Qiang Fu, Hao Song. On the minimum distances of binary optimal LCD codes with dimension 5[J]. AIMS Mathematics, 2024, 9(7): 19137-19153. doi: 10.3934/math.2024933
[1] | Takao Komatsu, Ram Krishna Pandey . On hypergeometric Cauchy numbers of higher grade. AIMS Mathematics, 2021, 6(7): 6630-6646. doi: 10.3934/math.2021390 |
[2] | Takao Komatsu, Huilin Zhu . Hypergeometric Euler numbers. AIMS Mathematics, 2020, 5(2): 1284-1303. doi: 10.3934/math.2020088 |
[3] | Nadia N. Li, Wenchang Chu . Explicit formulae for Bernoulli numbers. AIMS Mathematics, 2024, 9(10): 28170-28194. doi: 10.3934/math.20241366 |
[4] | Taekyun Kim, Hye Kyung Kim, Dae San Kim . Some identities on degenerate hyperbolic functions arising from p-adic integrals on Zp. AIMS Mathematics, 2023, 8(11): 25443-25453. doi: 10.3934/math.20231298 |
[5] | Jizhen Yang, Yunpeng Wang . Congruences involving generalized Catalan numbers and Bernoulli numbers. AIMS Mathematics, 2023, 8(10): 24331-24344. doi: 10.3934/math.20231240 |
[6] | Taekyun Kim, Dae San Kim, Jin-Woo Park . Degenerate r-truncated Stirling numbers. AIMS Mathematics, 2023, 8(11): 25957-25965. doi: 10.3934/math.20231322 |
[7] | Dojin Kim, Patcharee Wongsason, Jongkyum Kwon . Type 2 degenerate modified poly-Bernoulli polynomials arising from the degenerate poly-exponential functions. AIMS Mathematics, 2022, 7(6): 9716-9730. doi: 10.3934/math.2022541 |
[8] | Aimin Xu . Some identities connecting Stirling numbers, central factorial numbers and higher-order Bernoulli polynomials. AIMS Mathematics, 2025, 10(2): 3197-3206. doi: 10.3934/math.2025148 |
[9] | Waseem A. Khan, Abdulghani Muhyi, Rifaqat Ali, Khaled Ahmad Hassan Alzobydi, Manoj Singh, Praveen Agarwal . A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties. AIMS Mathematics, 2021, 6(11): 12680-12697. doi: 10.3934/math.2021731 |
[10] | Xhevat Z. Krasniqi . Approximation of functions in a certain Banach space by some generalized singular integrals. AIMS Mathematics, 2024, 9(2): 3386-3398. doi: 10.3934/math.2024166 |
Let da(n,5) and dl(n,5) be the minimum weights of optimal binary [n,5] linear codes and linear complementary dual (LCD) codes, respectively. This article aims to investigate dl(n,5) of some families of binary [n,5] LCD codes when n=31s+t≥14 with s integer and t∈{2,8,10,12,14,16,18}. By determining the defining vectors of optimal linear codes and discussing their reduced codes, we classify optimal linear codes and calculate their hull dimensions. Thus, the non-existence of these classes of binary [n,5,da(n,5)] LCD codes is verified, and we further derive that dl(n,5)=da(n,5)−1 for t≠16 and dl(n,5)=16s+6=da(n,5)−2 for t=16. Combining them with known results on optimal LCD codes, dl(n,5) of all [n,5] LCD codes are completely determined.
L. Carlitz ([1]) introduced analogues of Bernoulli numbers for the rational function (finite) field K=Fr(T), which are called Bernoulli-Carlitz numbers now. Bernoulli-Carlitz numbers have been studied since then (e.g., see [2,3,4,5,6]). According to the notations by Goss [7], Bernoulli-Carlitz numbers BCn are defined by
xeC(x)=∞∑n=0BCnΠ(n)xn. | (1.1) |
Here, eC(x) is the Carlitz exponential defined by
eC(x)=∞∑i=0xriDi, | (1.2) |
where Di=[i][i−1]r⋯[1]ri−1 (i≥1) with D0=1, and [i]=Tri−T. The Carlitz factorial Π(i) is defined by
Π(i)=m∏j=0Dcjj | (1.3) |
for a non-negative integer i with r-ary expansion:
i=m∑j=0cjrj(0≤cj<r). | (1.4) |
As analogues of the classical Cauchy numbers cn, Cauchy-Carlitz numbers CCn ([8]) are introduced as
xlogC(x)=∞∑n=0CCnΠ(n)xn. | (1.5) |
Here, logC(x) is the Carlitz logarithm defined by
logC(x)=∞∑i=0(−1)ixriLi, | (1.6) |
where Li=[i][i−1]⋯[1] (i≥1) with L0=1.
In [8], Bernoulli-Carlitz numbers and Cauchy-Carlitz numbers are expressed explicitly by using the Stirling-Carlitz numbers of the second kind and of the first kind, respectively. These properties are the extensions that Bernoulli numbers and Cauchy numbers are expressed explicitly by using the Stirling numbers of the second kind and of the first kind, respectively.
On the other hand, for N≥1, hypergeometric Bernoulli numbers BN,n ([9,10,11,12]) are defined by the generating function
11F1(1;N+1;x)=xN/N!ex−∑N−1n=0xn/n!=∞∑n=0BN,nxnn!, | (1.7) |
where
1F1(a;b;z)=∞∑n=0(a)(n)(b)(n)znn! |
is the confluent hypergeometric function with (x)(n)=x(x+1)⋯(x+n−1) (n≥1) and (x)(0)=1. When N=1, Bn=B1,n are classical Bernoulli numbers defined by
xex−1=∞∑n=0Bnxnn!. |
In addition, hypergeometric Cauchy numbers cN,n (see [13]) are defined by
12F1(1,N;N+1;−x)=(−1)N−1xN/Nlog(1+t)−∑N−1n=1(−1)n−1xn/n=∞∑n=0cN,nxnn!, | (1.8) |
where
2F1(a,b;c;z)=∞∑n=0(a)(n)(b)(n)(c)(n)znn! |
is the Gauss hypergeometric function. When N=1, cn=c1,n are classical Cauchy numbers defined by
xlog(1+x)=∞∑n=0cnxnn!. |
In [14], for N≥0, the truncated Bernoulli-Carlitz numbers BCN,n and the truncated Cauchy-Carlitz numbers CCN,n are defined by
xrN/DNeC(x)−∑N−1i=0xri/Di=∞∑n=0BCN,nΠ(n)xn | (1.9) |
and
(−1)NxrN/LNlogC(x)−∑N−1i=0(−1)ixri/Li=∞∑n=0CCN,nΠ(n)xn, | (1.10) |
respectively. When N=0, BCn=BC0,n and CCn=CC0,n are the original Bernoulli-Carlitz numbers and Cauchy-Carlitz numbers, respectively. These numbers BCN,n and CCN,n in (1.9) and (1.10) in function fields are analogues of hypergeometric Bernoulli numbers in (1.7) and hypergeometric Cauchy numbers in (1.8) in complex numbers, respectively. In [15], the truncated Euler polynomials are introduced and studied in complex numbers.
It is known that any real number α can be expressed uniquely as the simple continued fraction expansion:
α=a0+1a1+1a2+1a3+⋱, | (1.11) |
where a0 is an integer and a1,a2,… are positive integers. Though the expression is not unique, there exist general continued fraction expansions for real or complex numbers, and in general, analytic functions f(x):
f(x)=a0(x)+b1(x)a1(x)+b2(x)a2(x)+b3(x)a3(x)+⋱, | (1.12) |
where a0(x),a1(x),… and b1(x),b2(x),… are polynomials in x. In [16,17] several continued fraction expansions for non-exponential Bernoulli numbers are given. For example,
∞∑n=1B2n(4x)n=x1+12+x12+13+x13+14+x⋱. | (1.13) |
More general continued fractions expansions for analytic functions are recorded, for example, in [18]. In this paper, we shall give expressions for truncated Bernoulli-Carlitz numbers and truncated Cauchy-Carlitz numbers.
In [19], the hypergeometric Bernoulli numbers BN,n (N≥1, n≥1) can be expressed as
BN,n=(−1)nn!|N!(N+1)!10N!(N+2)!N!(N+1)!⋮⋮⋱10N!(N+n−1)!N!(N+n−2)!⋯N!(N+1)!1N!(N+n)!N!(N+n−1)!⋯N!(N+2)!N!(N+1)!|. |
When N=1, we have a determinant expression of Bernoulli numbers ([20,p.53]). In addition, relations between BN,n and BN−1,n are shown in [19].
In [21,22], the hypergeometric Cauchy numbers cN,n (N≥1, n≥1) can be expressed as
cN,n=n!|NN+110NN+2NN+1⋮⋮⋱10NN+n−1NN+n−2⋯NN+11NN+nNN+n−1⋯NN+2NN+1|. |
When N=1, we have a determinant expression of Cauchy numbers ([20,p.50]).
Recently, in ([23]) the truncated Euler-Carlitz numbers ECN,n (N≥0), introduced as
xq2N/D2NCoshC(x)−∑N−1i=0xq2i/D2i=∞∑n=0ECN,nΠ(n)xn, |
are shown to have some determinant expressions. When N=0, ECn=EC0,n are the Euler-Carlitz numbers, denoted by
xCoshC(x)=∞∑n=0ECnΠ(n)xn, |
where
CoshC(x)=∞∑i=0xq2iD2i |
is the Carlitz hyperbolic cosine. This reminds us that the hypergeometric Euler numbers EN,n ([24]), defined by
t2N/(2N)!cosht−∑N−1n=0t2n/(2n)!=∞∑n=0EN,nxnn!, |
have a determinant expression [25,Theorem 2.3] for N≥0 and n≥1,
EN,2n=(−1)n(2n)!|(2N)!(2N+2)!10(2N)!(2N+4)!⋱⋱0⋮⋱1(2N)!(2N+2n)!⋯(2N)!(2N+4)!(2N)!(2N+2)!|. |
When N=0, we have a determinant expression of Euler numbers (cf. [20,p.52]). More general cases are studied in [26].
In this paper, we also give similar determinant expressions of truncated Bernoulli-Carlitz numbers and truncated Cauchy-Carlitz numbers as natural extensions of those of hypergeometric numbers.
Let the n-th convergent of the continued fraction expansion of (1.12) be
Pn(x)Qn(x)=a0(x)+b1(x)a1(x)+b2(x)a2(x)+⋱+bn(x)an(x). | (2.1) |
There exist the fundamental recurrence formulas:
Pn(x)=an(x)Pn−1(x)+bn(x)Pn−2(x)(n≥1),Qn(x)=an(x)Qn−1(x)+bn(x)Qn−2(x)(n≥1), | (2.2) |
with P−1(x)=1, Q−1(x)=0, P0(x)=a0(x) and Q0(x)=1.
From the definition in (1.9), truncated Bernoulli-Carlitz numbers satisfy the relation
(DN∞∑i=0xrN+i−rNDN+i)(∞∑n=0BCN,nΠ(n)xn)=1. |
Thus,
Pm(x)=DN+mDN,Qm(x)=DN+mm∑i=0xrN+i−rNDN+i |
yield that
Qm(x)∞∑n=0BCN,nΠ(n)xn∼Pm(x)(m→∞). |
Notice that the n-th convergent pn/qn of the simple continued fraction (1.11) of a real number α yields the approximation property
|qnα−pn|<1qn+1. |
Now,
P0(x)Q0(x)=1=11,P1(x)Q1(x)=1−xrN+1−rNDN+1/DN+xrN+1−rN |
and Pn(x) and Qn(x) (n≥2) satisfy the recurrence relations
Pn(x)=(DN+nDN+n−1+xrN+n−rN+n−1)Pn−1(x)−DN+n−1DN+n−2xrN+n−rN+n−1Pn−2(x)Qn(x)=(DN+nDN+n−1+xrN+n−rN+n−1)Qn−1(x)−DN+n−1DN+n−2xrN+n−rN+n−1Qn−2(x) |
(They are proved by induction). Since by (2.2) for n≥2
an(x)=DN+nDN+n−1+xrN+n−rN+n−1andbn(x)=−DN+n−1DN+n−2xrN+n−rN+n−1, |
we have the following continued fraction expansion.
Theorem 1.
∞∑n=0BCN,nΠ(n)xn=1−xrN+1−rNDN+1DN+xrN+1−rN−DN+1DNxrN+2−rN+1DN+2DN+1+xrN+2−rN+1−DN+2DN+1xrN+3−rN+2DN+3DN+2+xrN+3−rN+2−⋱. |
Put N=0 in Theorem 1 to illustrate a simpler case. Then, we have a continued fraction expansion concerning the original Bernoulli-Carlitz numbers.
Corollary 1.
∞∑n=0BCnΠ(n)xn=1−xr−1D1+xr−1−D1xr2−rD2D1+xr2−r−D2D1xr3−r2D3D2+xr3−r2−⋱. |
From the definition in (1.10), truncated Cauchy-Carlitz numbers satisfy the relation
(LN∞∑i=0(−1)ixrN+i−rNLN+i)(∞∑n=0CCN,nΠ(n)xn)=1. |
Thus,
Pm(x)=LN+mLN,Qm(x)=LN+mm∑i=0(−1)ixrN+i−rNLN+i |
yield that
Qm(x)∞∑n=0CCN,nΠ(n)xn∼Pm(x)(m→∞). |
Now,
P0(x)Q0(x)=1=11,P1(x)Q1(x)=1+xrN+1−rNLN+1/LN−xrN+1−rN |
and Pn(x) and Qn(x) (n≥2) satisfy the recurrence relations
Pn(x)=(LN+nLN+n−1−xrN+n−rN+n−1)Pn−1(x)+LN+n−1LN+n−2xrN+n−rN+n−1Pn−2(x)Qn(x)=(LN+nLN+n−1−xrN+n−rN+n−1)Qn−1(x)+LN+n−1LN+n−2xrN+n−rN+n−1Qn−2(x). |
Since by (2.2) for n≥2
an(x)=LN+nLN+n−1−xrN+n+rN+n−1andbn(x)=LN+n−1LN+n−2xrN+n−rN+n−1, |
we have the following continued fraction expansion.
Theorem 2.
∞∑n=0CCN,nΠ(n)xn=1+xrN+1−rNLN+1LN−xrN+1−rN+LN+1LNxrN+2−rN+1LN+2LN+1−xrN+2−rN+1+LN+2LN+1xrN+3−rN+2LN+3LN+2−xrN+3−rN+2+⋱. |
In [14], some expressions of truncated Cauchy-Carlitz numbers have been shown. One of them is for integers N≥0 and n≥1,
CCN,n=Π(n)n∑k=1(−LN)k∑i1,…,ik≥1rN+i1+⋯+rN+ik=n+krN(−1)i1+⋯+ikLN+i1⋯LN+ik | (3.1) |
[14,Theorem 2].
Now, we give a determinant expression of truncated Cauchy-Carlitz numbers.
Theorem 3. For integers N≥0 and n≥1,
CCN,n=Π(n)|−a110a2−a1⋱⋮⋱⋱0⋮−a11(−1)nan⋯⋯a2−a1|, |
where
al=(−1)iLNδ∗lLN+i(l≥1) |
with
δ∗l={1if l=rN+i−rN(i=0,1,…);0otherwise. | (3.2) |
We need the following Lemma in [27] in order to prove Theorem 3.
Lemma 1. Let {αn}n≥0 be a sequence with α0=1, and R(j) be a function independent of n. Then
αn=|R(1)10R(2)R(1)⋮⋮⋱10R(n−1)R(n−2)⋯R(1)1R(n)R(n−1)⋯R(2)R(1)|. | (3.3) |
if and only if
αn=n∑j=1(−1)j−1R(j)αn−j(n≥1) | (3.4) |
with α0=1.
Proof of Theorem 3. By the definition (1.10) with (1.6), we have
1=(∞∑i=0(−1)iLNLN+i)xrN+i−rN(∞∑m=0CCmΠ(m)xm)=(∞∑l=0alxl)(∞∑m=0CCmΠ(m)xm)=∞∑n=0∞∑l=0alCCn−lΠ(n−l)xn. |
Thus, for n≥1, we get
∞∑l=0alCCn−lΠ(n−l)=0. |
By Lemma 1, we have
CCnΠ(n)=−n∑l=1alCCn−lΠ(n−l)=n∑l=1(−1)l−1(−1)lalCCn−lΠ(n−l)=|−a110a2−a1⋱⋮⋱⋱0⋮−a11(−1)nan⋯⋯a2−a1|. |
Examples. When n=rN+1−rN,
CCrN+1−rNΠ(rN+1−rN)=|01⋮⋱01(−1)rN+1−rNarN+1−rN0⋯0|=(−1)rN+1−rN+1(−1)rN+1−rN(−1)2N+1LNLN+1=LNLN+1. |
Let n=rN+2−rN. For simplicity, put
ˉa=(−1)rN+1−rN(−1)2N+1LNLN+1,ˆa=(−1)rN+2−rN(−1)2N+2LNLN+2. |
Then by expanding at the first column, we have
CCrN+1−rNΠ(rN+1−rN)=|010⋮⋱0ˉa0⋱⋮0⋱ˆa0⋯0⏟rN+2−rN+1ˉa⋱010⋯0⏟rN+1−rN−1|=(−1)rN+1−rN+1ˉa|1⋱1ˉa⋱⋱⏟rN+1−rN−101⋱⋱ˉa⏟2⋱1⋯0⏟rN+1−rN−1|+(−1)rN+2−rN+1ˆa|10⋱ˉa⋱⋱⋱0ˉa1|. |
The second term is equal to
(−1)rN+2−rN+1(−1)rN+2−rNLNLN+2=−LNLN+2. |
The first term is
(−1)rN+1−rN+1ˉa|01ˉa⋱⋱⏟rN+2−2rN+1+rNˉa10⏟rN+1−rN−1|=(−1)2(rN+1−rN+1)ˉa2|1⋱1ˉa⋱⋱⏟rN+1−rN−101⋱⋱ˉa⏟2⋱1⋯0⏟rN+1−rN−1|=(−1)r(rN+1−rN+1)ˉar|01⋮⋱01ˉa0⋯0|⏟rN+1−rN=(−1)(r+1)(rN+1−rN+1)ˉar+1=(−1)(r+1)(rN+1−rN+1)(−1)(rN+1−rN)(r+1)(−1)r+1Lr+1NLr+1N+1=Lr+1NLr+1N+1. |
Therefore,
CCrN+1−rNΠ(rN+1−rN)=Lr+1NLr+1N+1−LNLN+2. |
From this procedure, it is also clear that CCN,n=0 if rN+1−rN∤n, since all the elements of one column (or row) become zero.
In [14], some expressions of truncated Bernoulli-Carlitz numbers have been shown. One of them is for integers N≥0 and n≥1,
BCN,n=Π(n)n∑k=1(−DN)k∑i1,…,ik≥1rN+i1+⋯+rN+ik=n+krN1DN+i1⋯DN+ik | (4.1) |
[14,Theorem 1].
Now, we give a determinant expression of truncated Bernoulli-Carlitz numbers.
Theorem 4. For integers N≥0 and n≥1,
BCN,n=Π(n)|−d110d2−d1⋱⋮⋱⋱0⋮−d11(−1)ndn⋯⋯d2−d1|, |
where
dl=DNδ∗lDN+i(l≥1) |
with δ∗l as in (3.2).
Proof. The proof is similar to that of Theorem 3, using (1.9) and (1.2).
Example. Let n=2(rN+1−rN). For convenience, put
ˉd=DNDN+1. |
Then, we have
BCN,2(rN+1−rN)Π(2(rN+1−rN))=|01⋮0ˉd⋱ˉd⏟rN+1−rN+110⋯0⏟rN+1−rN−1|=(−1)rN+1−rN+1|1⋱1ˉd⋱⋱⏟rN+1−rN−10ˉd1⋱10⏟rN+1−rN−1|=(−1)rN+1−rN+1ˉd|01⋱1ˉd0|⏟rN+1−rN=(−1)2(rN+1−rN+1)ˉd2|1⋱1|=D2ND2N+1. |
It is also clear that BCN,n=0 if rN+1−rN∤n.
We shall use Trudi's formula to obtain different explicit expressions and inversion relations for the numbers CCN,n and BCN,n.
Lemma 2. For a positive integer n, we have
|a1a00⋯a2a1⋱⋮⋮⋮⋱⋱0an−1⋯a1a0anan−1⋯a2a1|=∑t1+2t2+⋯+ntn=n(t1+⋯+tnt1,…,tn)(−a0)n−t1−⋯−tnat11at22⋯atnn, |
where (t1+⋯+tnt1,…,tn)=(t1+⋯+tn)!t1!⋯tn! are the multinomial coefficients.
This relation is known as Trudi's formula [28,Vol.3,p.214], [29] and the case a0=1 of this formula is known as Brioschi's formula [30], [28,Vol.3,pp.208–209].
In addition, there exists the following inversion formula (see, e.g. [27]), which is based upon the relation
n∑k=0(−1)n−kαkD(n−k)=0(n≥1). |
Lemma 3. If {αn}n≥0 is a sequence defined by α0=1 and
αn=|D(1)10D(2)⋱⋱0⋮⋱⋱1D(n)⋯D(2)D(1)|, then D(n)=|α110α2⋱⋱0⋮⋱⋱1αn⋯α2α1|. |
From Trudi's formula, it is possible to give the combinatorial expression
αn=∑t1+2t2+⋯+ntn=n(t1+⋯+tnt1,…,tn)(−1)n−t1−⋯−tnD(1)t1D(2)t2⋯D(n)tn. |
By applying these lemmata to Theorem 3 and Theorem 4, we obtain an explicit expression for the truncated Cauchy-Carlitz numbers and the truncated Bernoulli-Carlitz numbers.
Theorem 5. For integers N≥0 and n≥1, we have
CCN,n=Π(n)∑t1+2t2+⋯+ntn=n(t1+⋯+tnt1,…,tn)(−1)n−t2−t4−⋯−t2⌊n/2⌋at11⋯atnn, |
where an are given in Theorem 3.
Theorem 6. For integers N≥0 and n≥1, we have
BCN,n=Π(n)∑t1+2t2+⋯+ntn=n(t1+⋯+tnt1,…,tn)(−1)n−t2−t4−⋯−t2⌊n/2⌋dt11⋯dtnn, |
where dn are given in Theorem 4.
By applying the inversion relation in Lemma 3 to Theorem 3 and Theorem 4, we have the following.
Theorem 7. For integers N≥0 and n≥1, we have
an=(−1)n|CCN,1Π(1)10CCN,2Π(2)CCN,1Π(1)⋮⋮⋱10CCN,n−1Π(n−1)CCN,n−2Π(n−2)⋯CCN,1Π(1)1CCN,nΠ(n)CCN,n−1Π(n−1)⋯CCN,2Π(2)CCN,1Π(1)|, |
where an is given in Theorem 3.
Theorem 8. For integers N≥0 and n≥1, we have
dn=(−1)n|BCN,1Π(1)10BCN,2Π(2)BCN,1Π(1)⋮⋮⋱10BCN,n−1Π(n−1)BCN,n−2Π(n−2)⋯BCN,1Π(1)1BCN,nΠ(n)BCN,n−1Π(n−1)⋯BCN,2Π(2)BCN,1Π(1)|, |
where dn is given in Theorem 4.
We would like to thank the referees for their valuable comments.
The authors declare no conflict of interest.
[1] | W. C. Huffman, V. Pless, Fundamentals of error-correcting codes, New York: Cambridge University Press, 2003. https://doi.org/10.1017/CBO9780511807077 |
[2] | E. F. Assmus Jr., J. D. Key, Affine and projective planes, Discrete Math., 83 (1990), 161–187. https://doi.org/10.1016/0012-365X(90)90003-Z |
[3] |
L. Lu, R. Li, L. Guo, Q. Fu, Maximal entanglement entanglement-assisted quantum codes constructed from linear codes, Quantum Inf. Process., 14 (2015), 165–182. https://doi.org/10.1007/s11128-014-0830-y doi: 10.1007/s11128-014-0830-y
![]() |
[4] | J. L. Massey, Linear codes with complementary duals, Discrete Math., 106-107 (1992), 337–342. https://doi.org/10.1016/0012-365X(92)90563-U |
[5] | C. Carlet, S. Guilley, Complementary dual codes for countermeasures to side-channel attacks, In: R. Pinto, P. Rocha Malonek, P. Vettori, Coding theory and applications, CIM Series in Mathematical Sciences, Berlin: Springer Verlag, 3 (2015), 97–105. https://doi.org/10.1007/978-3-319-17296-5_9 |
[6] |
C. Carlet, S. Mesnager, C. Tang, Y. Qi, R. Pellikaan, Linear codes over Fq are equivalent to LCD codes for q>3, IEEE Trans. Inform. Theory, 64 (2018), 3010–3017. https://doi.org/10.1109/TIT.2018.2789347 doi: 10.1109/TIT.2018.2789347
![]() |
[7] |
L. Galvez, J. L. Kim, N. Lee, Y. G. Roe, B. S. Won, Some bounds on binary LCD codes, Cryptogr. Commun., 10 (2018), 719–728. https://doi.org/10.1007/s12095-017-0258-1 doi: 10.1007/s12095-017-0258-1
![]() |
[8] |
Q. Fu, R. Li, F. Fu, Y. Rao, On the construction of binary optimal LCD codes with short length, Int. J. Found. Comput. Sci., 30 (2019), 1237–1245. https://doi.org/10.1142/S0129054119500242 doi: 10.1142/S0129054119500242
![]() |
[9] |
M. Harada, K. Saito, Binary linear complementary dual codes, Cryptogr. Commun., 11 (2019), 677–696. https://doi.org/10.1007/s12095-018-0319-0 doi: 10.1007/s12095-018-0319-0
![]() |
[10] |
M. Araya, M. Harada, K. Saito, Characterization and classification of optimal LCD codes, Des., Codes Cryptogr., 89 (2021), 617–640. https://doi.org/10.1007/s10623-020-00834-8 doi: 10.1007/s10623-020-00834-8
![]() |
[11] |
M. Araya, M. Harada, On the minimum weights of binary linear complementary dual codes, Cryptogr. Commun., 12 (2020), 285–300. https://doi.org/10.1007/s12095-019-00402-5 doi: 10.1007/s12095-019-00402-5
![]() |
[12] |
M. Araya, M. Harada, K. Saito, On the minimum weights of binary LCD codes and ternary LCD codes, Finite Fields Appl., 76 (2021), 101925. https://doi.org/10.1016/j.ffa.2021.101925 doi: 10.1016/j.ffa.2021.101925
![]() |
[13] | S. Bouyuklieva, Optimal binary LCD codes, Des. Codes Cryptogr., 89 (2021), 2445–2461. https://doi.org/10.1007/s10623-021-00929-w |
[14] | S. Li, M. Shi, Several constructions of optimal LCD codes over small finite fields, Cryptogr. Commun., 2024. https://doi.org/10.1007/s12095-024-00699-x |
[15] | R. Li, Y. Liu, Q. Fu, On some problems of LCD codes, 2022 Symposium on Coding Theory and Cryptography and Their Related Topics, Shandong: Zi Bo, 2022. |
[16] |
F. Li, Q. Yue, Y. Wu, Designed distances and parameters of new LCD BCH codes over finite fields, Cryptogr. Commun., 12 (2020), 147–163. https://doi.org/10.1007/s12095-019-00385-3 doi: 10.1007/s12095-019-00385-3
![]() |
[17] | W. C. Huffman, J. L. Kim, P. Solé, Concise encyclopedia of coding theory, Boca Raton: CRC Press, 2021,593–594. https://doi.org/10.1201/9781315147901 |
[18] | M. Grassl, Code tables: bounds on the parameters of various types of codes. Available from: http://www.codetables.de/. |
[19] |
R. Li, Z. Xu, X. Zhao, On the classification of binary optimal self-orthogonal codes, IEEE Trans. Inform. Theory, 54 (2008), 3778–3782. https://doi.org/10.1109/TIT.2008.926367 doi: 10.1109/TIT.2008.926367
![]() |
[20] | F. Zuo, R. Li, Y. Liu, Weight distributation of binary optimal codes and its application, 2012 International Conference on Computer Science and Information Processing (CSIP), Shaanxi: Xi'an, 2012,226–229. |
[21] |
J. E. MacDonald, Design methods for maximum minimum-distance error-correcting codes, IBM J. Res. Dev., 4 (1960), 43–57. https://doi.org/10.1147/rd.41.0043 doi: 10.1147/rd.41.0043
![]() |
[22] | The MathWorks, MATLAB R2006a, Natick, MA, 2006. Available from: https://blogs.mathworks.com/steve/2006/03/06/mathworks-product-release-r2006a/. |
[23] |
W. Bosma, J. Cannon, C. Playoust, The magma algebra system I: the user language, J. Symbolic Comput., 24 (1997), 235–265. https://doi.org/10.1006/jsco.1996.0125 doi: 10.1006/jsco.1996.0125
![]() |
[24] |
I. Bouyukliev, On the binary projective codes with dimension 6, Discrete Appl. Math., 154 (2006), 1693–1708. https://doi.org/10.1016/j.dam.2006.03.004 doi: 10.1016/j.dam.2006.03.004
![]() |