# Hermitian Self-Orthogonal Constacyclic Codes over Finite Fields

Authors: Amita Sahni; Poonam Trama Sehgal
Journal: Journal of Discrete Mathematics
 (2014)

---

## Abstract

Necessary and sufficient conditions for the existence of Hermitian self-orthogonal constacyclic codes of lengthn over a finite field Fq2, n coprime to q, are found. The defining sets and corresponding generator polynomials of these codes are also characterised. A formula for the number of Hermitian self-orthogonal constacyclic codes of length n over a finite field Fq2 is obtained. Conditions for the existence of numerous MDS Hermitian self-orthogonal constacyclic codes are obtained. The defining set and the number of such MDS codes are also found.


---

## Body

## 1. Introduction

LetFq2 denote a finite field with q2 elements. An [n,k] linear code C of length n and dimension k over Fq2 is a k-dimensional subspace of the vector space Fq2n. Elements of the subspace C are called codewords and are written as row vectors c=(c0,c1,…,cn-1). A linear code C over Fq2 is called λ-constacyclic if (λcn-1,c0,…,cn-2) is in C for every (c0,c1,…,cn-1) in C. Let θ:Fq2n→Fq2x/xn-λ be the map given by θc0,c1,…,cn-1↦c0+c1x+⋯+cn-1xn-1mod⁡(xn-λ). One can easily check that θ is an Fq2-module isomorphism. We can therefore identify λ-constacyclic codes of length n over Fq2 with ideals in Fq2[x]/xn-λ. The Hamming weight w(c) of c∈Fq2n is the number of nonzero coordinates of c. The minimum distance of C is defined to be d=min⁡⁡{w(c);c(≠0)∈C}. An [n,k,d] code, that is, a [n,k] linear code with minimum distance d, is said to be maximum distance separable (MDS) if d=n-k+1. The Hermitian inner product of elements u,v∈Fq2n is defined as u,vH=∑i=0n-1uiviq, for u=(u0,u1,…,un-1) and v=(v0,v1,…,vn-1). For a linear code C of length n over Fq2, the Hermitian dual code C⊥H of C is defined by C⊥H={v∈Fq2;u,vH=0,∀u∈C}. If C=C⊥H, then C is known as Hermitian self-dual and C is Hermitian self-orthogonal if C⊆C⊥H.Aydin et al. [1] dealt with constacyclic codes and a constacyclic BCH bound was given. Gulliver et al. [2] showed that there exists Euclidean self-dual MDS code of length q over Fq when q=2m by using a Reed-Solomon (RS) code and its extension. They also constructed many new Euclidean and Hermitian self-dual MDS codes over finite fields. Blackford [3] studied negacyclic codes over finite fields by using multipliers. He gave conditions on the existence of Euclidean self-dual codes. Recently, Guenda [4] constructed MDS Euclidean and Hermitian self-dual codes from extended cyclic duadic or negacyclic codes and gave necessary and sufficient conditions on the existence of Hermitian self-dual negacyclic codes arising from negacyclic codes. In [5] the authors gave formulae to enumerate the number of Euclidean self-dual and self-orthogonal negacyclic codes of length n over a finite field Fq, where q is coprime to n. In [6] Yang and Cai gave the necessary and sufficient conditions for the existence of Hermitian self-dual constacyclic codes. They also gave some conditions under which Hermitian self-dual and self-orthogonal MDS codes exist. In this paper, we find necessary and sufficient conditions for the existence of Hermitian self-orthogonal constacyclic codes of length n over a finite field Fq2, n coprime to q, and also give a characterization of their defining sets. We obtain a formula to calculate the number of these codes. We give conditions for the existence of some MDS Hermitian self-orthogonal constacyclic codes. We also found their number and defining sets (Table 1).Table 1
Number of Hermitian self-orthogonal codes overF25.
r
n
N
|
Λ
1
(
1
)
|
2
2
2
3
4
4
9
6
6
27
8
4
9
11
2
3
12
12
729
13
6
27
14
6
27
16
4
9
17
2
3
18
10
243
19
2
3
22
6
27
24
12
729
3
4
2
3
8
6
27
11
2
3
12
2
3
13
6
27
16
10
243
17
2
3
19
2
3
22
4
9
24
6
27
26
12
729
28
6
27
31
10
243
32
14
2187
6
2
2
3
4
4
9
6
2
3
8
4
9
11
2
3
12
4
9
13
6
27
14
6
27
16
4
9
17
2
3
18
2
3
19
2
3
22
6
27
24
4
9


## 2. Hermitian Self-Orthogonal Constacyclic Codes

Letq be an odd prime power and n a positive integer relatively prime to q. Let C be an [n,k]  λ-constacyclic code over Fq2 with r=ordq2(λ), where ordq2(λ) denotes the order of λ in Fq2. Let g(x) be the generator polynomial of C. Then g(x) divides (xn-λ). Write (xn-λ)=g(x)h(x). The polynomial h(x) is called the check polynomial of C. For 0≤s<rn, let Cs={s,sq2,…,s(q2)ns-1} be the q2-cyclotomic coset modulo rn containing s, where ns is the least positive integer such that s(q2)ns≡s(mod⁡rn). Let α be a primitive rn th root of unity in some extension field of Fq2 such that αn=λ. Then the polynomial M(s)(x)=∏i∈Cs(x-αi) is the minimal polynomial of αs over Fq2 and
(1)xrn-1=∏s∈Λ‍M(s)(x),
where Λ is the set of representatives of all the distinct q2-cyclotomic cosets modulo rn. As xn-λ∣(xrn-1), one can check that the roots of (xn-λ) are precisely αir+1, 0≤i<n. Define
(2)Or,n(1)={ir+1;0≤i<n}(mod⁡rn).
Hence we have
(3)xn-λ=∏s∈Λ1‍Msx,
where Λ1=Λ∩Or,n(1).LetC=g(x) be a λ-constacyclic code with defining set T={ir+1∈Or,n(1);αir+1 is a root of g(x)}. Clearly T is a union of some q2-cyclotomic cosets Cs modrn for s∈Λ1. The Hermitian dual C⊥H of the code C is a λ-q-constacyclic code over Fq2 with defining set T⊥H=-q[Or,n(1)∖T] modrn (see Theorem 3.2 of [6]). Write the generator polynomial g(x) of the code C as g(x)=∏s∈Λ1M(s)(x)δ(s), where
(4)δ(s)=1,ifs∈T,0,ifs∉T.
Then the generator polynomial of the Hermitian dual C⊥H of C is h⊥H(x)=∏s∈Λ1(M(s)(x))Δ(s), where
(5)Δ(s)=1,ifs∈T⊥H,0,ifs∉T⊥H.
It can be easily verified that Δ(s)=1-δ(-qs) so that h⊥H(x)=∏s∈Λ1M(s)(x)1-δ(-qs).Lemma 1.
LetC be a λ-constacyclic code over Fq2 with ordq2⁡(λ)=r. If C is a Hermitian self-orthogonal code, then r∣(q+1).Proof.
The proof is similar to [6, Proposition 2.3].Theorem 2.
Nontrivial Hermitian self-orthogonalλ-constacyclic codes of length n over Fq2 exist if and only if Cs≠C-qs for some s∈Or,n(1).Proof.
LetC be a nontrivial Hermitian self-orthogonal λ-constacyclic code of length n over Fq2 with defining set T. Then there exists s∈Or,n(1) such that s≠0 and s∉T. Hence s∈Or,n(1)∖T giving us that -qs∈T⊥H⊆T. Thus, Cs≠C-qs (as s∉T and -qs∈T). Conversely, let s∈Or,n(1) be such that Cs≠C-qs. Consider T=Or,n(1)∖Cs. The code C is a nontrivial Hermitian self-orthogonal code since T⊥H=-q[Or,n(1)∖T]=-qCs=C-qs⊆T.DefineT0={s∈Or,n(1);Cs=C-qs}. The following theorem characterizes the defining set of a Hermitian self-orthogonal constacyclic code of length n over Fq2.Theorem 3.
LetC be a λ-constacyclic code of length n over Fq2 with the defining set T. Then C is Hermitian self-orthogonal if and only if (i) T0⊂T and (ii) for each s∉T0, at least one of s and -qs belongs to T.Proof.
LetC be a Hermitian self-orthogonal constacyclic code. Let s∈T0. Then Cs=C-qs. Suppose that s∉T. Then -qs∈T⊥H⊆T so that C-qs⊂T and Cs⊈T, which contradicts the hypothesis that Cs=C-qs. Thus, T0⊆T. Now, let s∉T0. Then either s∈T or -qs∈T⊥H⊆T, as required.
Conversely, let the defining setT be such that T0⊂T and for each s∉T0, at least one of s and -qs is in T. Then T⊥H=-q[Or,n(1)∖T]={-qs;s∉T}⊆T, by condition (ii) so that the code C having T as a defining set is a Hermitian self-orthogonal code.Corollary 4.
Aλ-constacyclic code C of length n over Fq2 generated by g(x)=∏s∈Λ1M(s)(x)δ(s) is a Hermitian self-orthogonal if and only if δ(s)+δ(-qs)≥1 for all s∈Λ1.DefineΛ1(0)=Λ1∩T0 and Λ1(1)=Λ1∖Λ1(0). Observe that T0=⋃s∈Λ1(0)Cs.Example 5.
Letq=13, n=15, and r=7; then q2=169. We consider the λ-constacyclic code of length 15 over F132, where λ∈F132 with order 7. Clearly
(6)O7,15(1)={1,8,15,22,29,36,43,50,57,64,71,78,85,92,99},T0={15}. Let T={1,15,22,36,43,50,64,99}; then T⊥H=-13[O7,15(1)∖T]={1,22,36,43,50,64,99}⊂T. Hence, the code with defining set T is a [15,7] Hermitian self-orthogonal λ-constacyclic code.Theorem 6.
The number of Hermitian self-orthogonalλ-constacyclic codes of length n over Fq2 is 3Λ1(1)/2.Proof.
LetC be a Hermitian self-orthogonal λ-constacyclic code of length n over Fq2 generated by g(x)=∏s∈Λ1M(s)(x)δ(s). Then δ(s)+δ(-qs)≥1. For s∈Λ1(0), δ(s)=δ(-qs)=1. However for s∈Λ1(1), the pairs (δ(s),δ(-qs)) have three choices (0,1), (1,0), and (1,1). Hence, the number of Hermitian self-orthogonal λ-constacyclic codes of length n over Fq2 is 3Λ1(1)/2.In order to find the number of Hermitian self-orthogonalλ-constacyclic codes of length n over Fq2, we need to compute the value of Λ1(1). Our aim is to prove the following.Theorem 7.
Leta, r, and d be positive integers such that gcd(a,r)=1. Then the number of solutions for the linear congruence
(7)ax≡1(mod⁡r)
in the set A={w;0≤w<rd and gcd(w,rd)=1} is exactly ϕ(rd)/ϕ(r).Sincegcd(a,r)=1, the linear congruence ax≡1(mod⁡r) has a unique solution modulo r. Let it be x≡b(mod⁡r). Then gcd(b,r)=1. We write A={ir+s;0≤i<d,0≤s<r,gcd(ir+s,rd)=1}. The solutions of (7) will be amongst Ab={ir+b;0≤i<d}. Clearly, the elements of Ab are relatively prime to r. We need to count the number of elements of Ab which are coprime to d. Also, Ab=d. Therefore, the required number N=d-{ir+b;0≤i<d,gcd(ir+b,d)>1}.Lemma 8.
Letp be a prime divisor of d such that p∤r. The number of multiples of p in Ab is d/p.Proof.
WriteAb=⋃k=0(d/p)-1Ab(k), where Ab(k)={kpr+b,(kp+1)r+b,…,((k+1)p-1)r+b} for each k, 0≤k<d/p. Since each Ab(k) contains p elements which are pairwise incongruent mod p, each Ab(k) forms a complete residue system mod p. Hence exactly one element in each Ab(k) is divisible by p. Consequently, there exist d/p elements in Ab which are divisible by p.Lemma 9.
Letp1 and p2 be two distinct prime divisors of d with gcd(p1p2,r)=1. Then the number of multiples of p1 or p2 in Ab is d/p1+d/p2-d/p1p2.Proof.
The number of multiples ofp1 in Ab equals d/p1, while the number of multiples of p2 in Ab is d/p2. By a similar argument as in Lemma 8, the number of multiples of both p1 and p2 equals d/p1p2. Therefore the required number is d/p1+d/p2-d/p1p2.Theorem 10.
Letp1,p2,…,pm be all the distinct prime divisors of d which are relatively prime to r. The number of elements in Ab which are not coprime to d is
(8)N0=∑1≤i≤mdpi-∑1≤i,j≤mi≠jdpipj+∑1≤i,j,k≤mi,j,kdistinctdpipjpk+⋯+-1m-1dp1p2…pm.Proof.
The proof follows by induction onm Lemmas 8 and 9.In order to prove Theorem7, it is enough to show that N=d-N0=ϕ(rd)/ϕ(r).Now,(9)N=d-∑1≤i≤mdpi-∑1≤i,j≤mi≠jdpipj+∑1≤i,j,k≤mi,j,kdistinctdpipjpkwwwww+⋯+-1m-1dp1p2⋯pm∑1≤i,j,k≤mi,j,kdistinctdpipjpk=d1-1p11-1p2⋯1-1pm.Letp1,p2,…,pm+l be all the distinct prime divisors of d. Also, let pm+1,…,pm+l,pm+l+1,…,pm+l+k be all the distinct prime divisors of r. Then (d-N0)ϕ(r)=dr∏i=1m+l+k1-(1/pi)=ϕ(rd), which completes the proof of Theorem 7.Picks∈Λ1. Let gcd(rn,s)=ms. Define Λ1,ms={s∈Λ1;gcd(rn,s)=ms}.Theorem 11.
Consider(10)Λ1,ms=ϕ(rds)ordrdsq2ϕr,ifgcd(ms,r)=1,0,ifgcd(ms,r)≠1,
where ds=n/ms.Proof.
Ass∈Λ1, s≡1(mod⁡r), so that gcd(r,s)=1. Since ms is a divisor of s, we have that gcd(ms,r)=1. Thus Λ1,ms=0, whenever ms is not coprime to r.
Asgcd(rn,s)=gcd(n,s)=ms, we write s=mss′. Also gcd(ds,s′)=gcd(n/ms,s/ms)=1. As s∈Λ1, s≡1(mod⁡r) holds giving us that mss′≡1(mod⁡r) holds with gcd(ds,s′)=1 and 0≤s′<rds (∵0≤s<rn). By Theorem 7, there is ϕ(rds)/ϕ(r) number of elements s=mss′ in Or,n(1) satisfying 0≤s′<rds, gcd(ds,s′)=1, and mss′≡1(mod⁡r). However, we have to calculate the number of such s in Λ1. Now, for s∈Λ1, gcd(rn,s)=ms,
(11)Cs=ordrn/ms(q2)=ordrds(q2).
Consequently, Λ1,ms=ϕ(rds)/ordrds(q2)ϕ(r), whenever gcd(ms,r)=1.Define(12)χ(d,q)=0,ifqk≡-1(mod⁡d)forsomeoddintegerk,1,otherwise.
Observe that, for s∈Λ1, Cs=C-qs holds if and only if χ(rds,q)=0, where ds=n/ms, ms=gcd(s,n) as defined earlier. Recall that, for s∈Λ1, gcd(ms,r)=1.Theorem 12.
One has(13)Λ1(1)=∑d∣ngcd(n/d,r)=1χ(rd,q)ϕ(rd)ϕrordrd⁡(q2).Proof.
The proof follows from Theorem11 and the above definition.Example 13 (letn=12, r=3, and q=5).
ThenOr,n(1)={1,4,7,10,13,16,19,22,25,28,31,34}, C1={1,25,13}, C4={4,28,16}, C7={7,31,19}, and C10={10,34,22}. Take Λ1={1,4,7,10}. Thus Λ1(1)={1,7} (as C4=C-20 and C10=C-50) so that Λ1(1)=2. By Theorem 12, d=3,6,12 are possible values of d on right hand side. Now, χ(9,5)=χ(18,5)=0 as 53≡-1(mod⁡18). However, χ(36,5)=1 as there does not exist any odd integer k such that 5k≡-1(mod⁡36), so that
(14)Λ1(1)=χ(36,5)ϕ(36)ϕ(3)ord36(25)=123×2=2.We will now investigate the behavior of the functionχ(d,q).Lemma 14.
Letm1 and m2 be two integers coprime to q such that qk1≡-1(mod⁡m1) and qk2≡-1(mod⁡m2) for some odd integers k1 and k2. If q≡-1(mod⁡2a+b), where 2a∥m1 and 2b∥m2, then there exists an odd integer k such that qk≡-1(mod⁡m1m2).Proof.
Writem1m2=2a+b∏i=1mpiei, pi being odd distinct primes, ei≥1. Let p be an odd prime divisor of m1m2, so that there exists an odd integer t(=k1ork2) such that qt≡-1(mod⁡p). Therefore, q2t≡1(mod⁡p). Consequently, ordpq∣2t but ordpq∤t, giving us that 2∥ordp(q). For e≥1 we have 2∥ordpe(q), since ordpe(q)=pfordp(q) for some positive integer f. Hence qkp,e≡-1(mod⁡pe), where kp,e=ordpe(q)/2 is odd. Thus,
(15)qk≡-1(mod⁡m1m2),
where k=lcm{kpi,ei}i=1m is odd.Lemma 15.
Leta≥2. Then qk≡-1(mod⁡2a) holds for some integer k≥1 if and only if q≡-1(mod⁡2a). In fact, such a k is odd.Proof.
Proof is trivial.Theorem 16.
χ
(
r
d
,
q
)
=
0 if and only if q≡-1(mod⁡2a+b) and 2∥ordp(q) for all odd prime divisors p of d, where 2a∥d and 2b∥r.Proof.
Ifχ(rd,q)=0, then there exists an odd integer k such that qk≡-1(mod⁡rd). Thus qk≡-1(mod⁡2a+b) so that, by Lemma 15, q≡-1(mod⁡2a+b). Also qk≡-1(mod⁡p) for every odd prime divisor p of d. Thus, q2k≡1(mod⁡p) showing that ordp(q)|2k and ordp(q)∤k. Therefore 2∥ordp(q) for all odd prime divisors p of d.
Conversely, letq≡-1(mod⁡2a+b) and 2∥ordp(q) for all odd prime divisors p of d. To prove χ(rd,q)=0, we need to find an odd integer k such that qk≡-1(mod⁡rd). For any odd prime divisor p of d, as 2∥ordp(q), qkp≡-1(mod⁡p), where kp=ordp(q)/2 is odd. As in the proof of Lemma 14, there exists an odd integer kd such that qkd≡-1(mod⁡d′), where d=2ad′. Also, q≡-1(mod⁡2a). Therefore, qkd≡-1(mod⁡d) with kd odd. By Lemma 1, q≡-1(mod⁡r). Using Lemma 14, we get that qk≡-1(mod⁡rd), for some odd integer k.Proposition 17.
Ifr and n are coprime, then χ(rd,q)=χ(d,q) for all divisors d of n.Proof.
Sinceq≡-1(mod⁡r) and gcd(r,d)=1, for some odd integer k, qk≡-1(mod⁡rd) if and only if qk≡-1(mod⁡d).Corollary 18.
There does not exist any nontrivial Hermitian self-orthogonalλ-constacyclic code of length n over Fq2 if and only if q≡-1(mod⁡2a+b) and 2∥ordp(q) for all odd prime divisors p of n, where 2a∥n and 2b∥r, for r=ordq2(λ).Proof.
The proof follows easily from Theorems12 and 16.Forn odd, we have a=0. The condition q≡-1(mod⁡2a+b) reads as q≡-1(mod⁡2b), which is always true as r∣(q+1). Hence, we have the following.Corollary 19.
There does not exist any nontrivial Hermitian self-orthogonalλ-constacyclic code of odd length n over Fq2 if and only if 2∥ordp(q) for all prime divisors p of n.


## 3. MDS Hermitian Self-Orthogonal Constacyclic Codes OverFq2

LetC be a λ-constacyclic code of length n over Fq2 and ordq2(λ)=r. Let α be a primitive rn th root of unity in some extension field of Fq2 such that αn=λ. Then roots of C are of the form αir+1, 0≤i≤n-1. Put ζ=αr.Theorem 20.
Let the generator polynomial ofC have roots that include the set {αζi;i1+1≤i≤i1+d-1}. Then the minimum distance of C is at least d.Proof.
See [1, Theorem 2.2]By Lemma1, r∣(q+1). Write q+1=rs0.Theorem 21.
Letn be a divisor of (q-1). Let T=Or,n(1)∖Tl,m, where Tl,m={ir+1;l≤i≤m}(mod⁡rn) for each m≤(n-1-s0)/2 and each l≥l0, with
(16)l0=-s0-12,ifniseven,-s0-22,ifnisodd.
Then the code C with defining set T is a Hermitian self-orthogonal λ-constacyclic MDS code with parameters n,m-l+1,n-m+l.Proof.
Letl, m be as above and
(17)Tl,m={ir+1;l≤i≤m}(mod⁡rn).
Each Tl,m has m-l+1 elements. If C denotes the code with defining set T=Or,n(1)∖Tl,m, then the dimension of C, dim⁡⁡C=m-l+1. Let I={i;l≤i≤m}. Then the set I^={0,1,…(n-1)}∖I has n-(m-l+1)=n-m+l-1 consecutive elements modulo n. By Theorem 20, the minimum distance of C is at least (n-m+l). However, using the singleton bound, the minimum distance is at most (n-m+l). Consequently, the minimum distance of C equals (n-m+l), proving that C is an MDS code.
In order to prove thatC is self-orthogonal, it is enough to prove that Tl,m∩(-qTl,m)=∅. We have -q(ir+1)≡(n-i-s0)r+1(mod⁡rn). Let I1={i;l≤i≤m}(mod⁡n) and I2={n-i-s0;i∈I1}(mod⁡n). Then Tl,m∩(-qTl,m)=∅ if and only if I1∩I2=∅. Let i∈I1∩I2, then i=n-j-s0 for some j∈I1 so that
(18)s0+i+j=n≡0(mod⁡n).
As i,j∈I1, l0≤l≤i, j≤m≤(n-s0-1)/2. Thus -s0<2l0≤2l≤i+j≤2(n-s0-1)/2≤n-s0-1<n-s0 so that
(19)0<s0+i+j<n.
Consequently, there does not exist any i, j such that both (18) and (19) hold, thereby showing that C is a Hermitian self-orthogonal constacyclic MDS code.Remark 22.
Forn even, l=l0=-(s0-1)/2 and m=(n-s0-1)/2, the codes obtained from above theorem are the same as given by Theorem 4.3 of [6].Proposition 23.
LetN be the number of Hermitian self-orthogonal λ-constacyclic MDS codes of length n which can be obtained from above theorem. Then N is given by
(20)N=n(n-2)8,ifsiseven,neven,n(n+2)8,ifsisodd,neven,(n-1)(n+1)8,ifsiseven,nodd,(n-3)(n-1)8,ifsisodd,nodd.Proof.
The required numberN equals the number of Tl,m={ir+1;l≤i≤m}(mod⁡rn); that is, we need to select m-l+1 consecutive integers from the set {i;l0≤i≤m0=(n-1-s0)/2}. The number of ways of selecting t consecutive integers from the set {i;e≤i≤f} is (f-e-t+2). Thus, the number of such codes is N=∑j=1m0-l0+1j=(m0-l0+1)(m0-l0+2)/2. Hence the result follows.Tables2 and 3 list [n,k,d] Hermitian self-orthogonal MDS codes over Fq2 for q≤11. Here n, k, and d denote, respectively, the length, dimension, and minimum distance of the code while T denotes the defining set. N denotes the number of such Hermitian self-orthogonal MDS codes.Table 2
[
n
,
k
,
d
] Hermitian self-orthogonal MDS codes over Fq2.
q
n
r
N
k
d
T
5
4
1
1
1
4
{
0,1
,
2
}
5
4
2
3
2
3
{
3,5
}
1
4
{
1
,
3,5
}
1
4
{
3,5
,
7
}
5
4
3
1
1
4
{
4,7
,
10
}
5
4
6
3
2
3
{
13,19
}
1
4
{
7,13,19
}
1
4
{
1,13,19
}
7
3
1
1
1
3
{
0,2
}
7
3
2
1
1
3
{
1,3
}
7
3
4
1
1
3
{
5,9
}
7
6
1
3
2
5
{
0,1
,
2,3
}
1
6
{
0,1
,
2,3
,
4
}
1
6
{
0,1
,
2,3
,
5
}
7
6
2
3
2
5
{
3,5
,
7,9
}
1
6
{
1,3
,
5,7
,
9
}
1
6
{
3,5
,
7,9
,
11
}
7
6
4
3
2
5
{
9,13,17,21
}
1
6
{
1,9
,
13,17,21
}
1
6
{
5,9
,
13,17,21
}
7
6
8
6
3
4
{
25,33,41
}
2
5
{
17,25,33,41
}
2
5
{
1,25,33,41
}
1
6
{
9,17,25,33,41
}
1
6
{
1,17,25,33,41
}
1
6
{
1,9
,
25,33,41
}
11
5
1
3
2
4
{
0,3
,
4
}
1
5
{
0,2
,
3,4
}
1
5
{
0,1
,
3,4
}
11
5
2
3
2
4
{
1,3
,
5
}
1
5
{
1,3
,
5,9
}
1
5
{
1,3
,
5,7
}
11
5
3
3
2
4
{
4,7
,
10
}
1
5
{
1,4
,
7,10
}
1
5
{
4,7
,
10,13
}
11
5
4
1
1
5
{
5,9
,
13,17
}
11
5
6
3
2
4
{
13,19,25
}
1
5
{
7,13,19,25
}
1
5
{
1,13,19,25
}
11
5
12
1
1
5
{
25,37,49,1
}
11
10
1
10
4
7
{
0,1
,
2,3
,
4,5
}
3
8
{
0,1
,
2,3
,
4,5
,
9
}
3
8
{
0,1
,
2,3
,
4,5
,
6
}
2
9
{
0,1
,
2,3
,
4,5
,
8,9
}
2
9
{
0,1
,
2,3
,
4,5
,
6,9
}
2
9
{
0,1
,
2,3
,
4,5
,
6,7
}
1
10
{
0,1
,
2,3
,
4,5
,
7,8
,
9
}
1
10
{
0,1
,
2,3
,
4,5
,
6,8
,
9
}
1
10
{
0,1
,
2,3
,
4,5
,
6,7
,
9
}
1
10
{
0,1
,
2,3
,
4,5
,
6,7
,
8
}
11
10
2
10
4
7
{
5,7
,
9,11,13,15
}
3
8
{
3,5
,
7,9
,
11,13,15
}
3
8
{
5,7
,
9,11,13,15,17
}
2
9
{
1,3
,
5,7
,
9,11,13,15
}
2
9
{
3,5
,
7,9
,
11,13,15,17
}
2
9
{
5,7
,
9,11,13,15,17,19
}
1
10
{
1,3
,
5,7
,
9,11,13,15,19
}
1
10
{
1,3
,
5,7
,
9,11,13,15,17
}
1
10
{
3,5
,
7,9
,
11,13,15,17,19
}
1
10
{
1,5
,
7,9
,
11,13,15,17,19
}Table 3
[
n
,
k
,
d
] Hermitian self-orthogonal MDS codes over  Fq2.
q
n
r
N
k
d
T
11
10
3
10
4
7
{
10,13,16,19,22,25
}
3
8
{
7,10,13,16,19,22,25
}
3
8
{
10,13,16,19,22,25,28
}
2
9
{
4,7
,
10,13,16,19,22,25
}
2
9
{
7,10,13,16,19,22,25,28
}
2
9
{
1,10,13,16,19,22,25,28
}
1
10
{
1,4
,
7,10,13,16,19,22,25
}
1
10
{
4,7
,
10,13,16,19,22,25,28
}
1
10
{
1,7
,
10,13,16,19,22,25,28
}
1
10
{
1,4
,
10,13,16,19,22,25,28
}
11
10
4
15
5
6
{
17,21,25,29,33
}
4
7
{
13,17,21,25,29,33
}
4
7
{
17,21,25,29,33,37
}
3
8
{
9,13,17,21,25,29,33
}
3
8
{
13,17,21,25,29,33,37
}
3
8
{
1,17,21,25,29,33,37
}
2
9
{
5,9
,
13,17,21,25,29,33
}
2
9
{
9,13,17,21,25,29,33,37
}
2
9
{
1,13,17,21,25,29,33,37
}
2
9
{
1,5
,
17,21,25,29,33,37
}
1
10
{
1,5
,
9,13,17,21,25,29,33
}
1
10
{
5,9
,
13,17,21,25,29,33,37
}
1
10
{
1,9
,
1317,21,25,29,33,37
}
1
10
{
1,5
,
1317,21,25,29,33,37
}
1
10
{
1,5
,
9,17,21,25,29,33,37
}
11
10
6
10
4
7
{
25,31,37,43,49,55
}
3
8
{
19,25,31,37,43,49,55
}
3
8
{
1,25,31,37,43,49,55
}
2
9
{
13,19,25,31,37,43,49,55
}
2
9
{
1,19,25,31,37,43,49,55
}
2
9
{
1,7
,
25,31,37,43,49,55
}
1
10
{
7,13,19,25,31,37,43,49,55
}
1
10
{
1,13,19,25,31,37,43,49,55
}
1
10
{
1,7
,
19,25,31,37,43,49,55
}
1
10
{
1,7
,
13,25,31,37,43,49,55
}
11
10
12
15
5
6
{
61,73,85,97,109
}
4
7
{
49,61,73,85,97,109
}
4
7
{
1,61,73,85,97,109
}
3
8
{
37,49,61,73,85,97,109
}
3
8
{
1,49,61,73,85,97,109
}
3
8
{
1,13,61,73,85,97,109
}
2
9
{
25,37,49,61,73,85,97,109
}
2
9
{
1,37,49,61,73,85,97,109
}
2
9
{
1,13,49,61,73,85,97,109
}
2
9
{
1,13,25,61,73,85,97,109
}
1
10
{
13,25,37,49,61,73,85,97,109
}
1
10
{
1,25,37,49,61,73,85,97,109
}
1
10
{
1,13,37,49,61,73,85,97,109
}
1
10
{
1,13,25,49,61,73,85,97,109
}
1
10
{
1,13,25,3761,73,85,97,109
}



---

*Source: 985387-2014-11-12.xml*