Weight | Frequency |
0 | 1 |
8 | 40 |
10 | 80 |
11 | 32 |
12 | 80 |
16 | 10 |
We introduce a sequence of third and fourth-order iterative schemes for finding the roots of nonlinear equations by using the decomposition technique and Simpson's one-third rule. We also discuss the convergence analysis of our suggested iterative schemes. With the help of different numerical examples, we demonstrate the validity, efficiency and implementation of our proposed schemes.
Citation: Awais Gul Khan, Farah Ameen, Muhammad Uzair Awan, Kamsing Nonlaopon. Some new numerical schemes for finding the solutions to nonlinear equations[J]. AIMS Mathematics, 2022, 7(10): 18616-18631. doi: 10.3934/math.20221024
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We introduce a sequence of third and fourth-order iterative schemes for finding the roots of nonlinear equations by using the decomposition technique and Simpson's one-third rule. We also discuss the convergence analysis of our suggested iterative schemes. With the help of different numerical examples, we demonstrate the validity, efficiency and implementation of our proposed schemes.
Recently, several infinite families of minimal and optimal linear codes are constructed via mathematical objects named simplicial complexes or down-sets by Hyun and Wu et al [3,5,7,8,12,13]. Simplicial complexes are extremely well-behaved with the n-variable generating function, which in turn enable us to compute the exponential sum rather efficiently. Let n be a natural number and denote by [n]={1,2,…,n} the set of integers from 1 to n. For Δ⊆P([n]), we say Δ is a simplicial complex if u∈Δ and v⊆u imply v∈Δ, where P([n]) denotes the power set of [n]. The set-inclusion defines a partial order on Δ. A maximal element of a simplicial complex Δ is an element of Δ that is not smaller than any other element in Δ. For subsets Ai of [n], where i∈[S], the notation ⟨A1,A2,…,As⟩ means it is a simplicial complex generated by {A1,A2,…,As}, that is ⟨A1,A2,…,As⟩={B:B⊆Ai,i∈[S]}. Especially, when s=1, we write ⟨A1⟩ simply as ΔA1.
Ternary codes of small dimension have been investigated in many literatures, see for instance [2,6,9,10,11]. A class of group character ternary codes C3(1,n−1) with parameters [2n−1,n,2n−2], which are the analogue of the binary first-order Reed-Muller codes RM(1,n−1) are described and analyzed by Ding et al. [4]. In this paper, we describe a new class of [2n−1,n,2n−2] ternary codes, and determine their weigt distributions.
Minimal linear codes, though existing as special linear codes, have important applications in secret sharing and secure two-party computation. Construction of minimal linear codes with new and desirable parameters would be an interesting topic in coding theory and cryptography. We construct in this paper a family of minimal linear codes over F3, and compute their weight distributions. By a distance-optimal code, or simply an optimal code, we mean it has the highest minimum distance with a prescribed length and dimension. One class of these minimal codes we obtained is proved to be optimal.
In this paper we study a linear code with more flexible lengths as follows. Let P be a subset of Fn3, and we order the elements of P to fix a coordinate position of vectors. A ternary code CP associated with P is defined to be
CP={cP(u)=(u⋅x)x∈P:u∈Fn3}. |
It is straightforward that CP is a linear code of length |P| and its dimension is at most n.
For a subset P of Fn3 and u∈Fn3, we define the exponential sum with respect to P by
χu(P)=∑v∈Pζu⋅v, |
where ζ is a primitive 3-rd root of the unity. Then the Hamming weight of a codeword cP(u) in CP is given as follows:
w(cP(u))=|P|−∑v∈Pδ0,u⋅v=|P|−13∑y∈F3∑v∈Pζy(u⋅v)=|P|−13(|P|+2Re(∑v∈Pζu⋅v))=23(|P|−Re(χu(P))) | (2.1) |
where δ is the Kronecker delta function and Re(χu(P)) is the real part of χu(P). The main difficulty of the computation of w(cP(u)) lies in the fact that it is expressed as the exponential sum with respect to a subset P which in turn is hard to compute for an arbitrary P.
When P contains the zero-vector of Fn3, we are also interested in CPc where Pc denotes the complement of P, that is
CPc={cPc(u)=(u⋅x)x∈Pc:u∈Fn3}. |
Then the weight of cPc(u) and that of cP(u) are related as follows:
w(cPc(u))=2⋅3n−1(1−δ0,u)−w(cP(u)). | (2.2) |
For the purpose of computing the exponential sum χu(P), we introduce the following n-variable generating function associated with P inspired by Adamaszek [1]:
HP(x1,x2,…,xn)=∑v∈Pn∏i=1xvii∈Z[x±11,…,x±1n] |
where we denote v=(v1,v2,…,vn) if v∈Fn3. By convention, we define HP(x1,x2,…,xn)=0 if P=∅.
Example 1. Let P={(1,−1,−1,…,−1)}, then the generating function is
HP(x1,x2,…,xn)=x1x2x3⋯xn. |
In general, one can easily obtain the following result when P=(F∗3)n
HP(x1,x2,…,xn)=1x1x2⋯xnn∏i=1(1+x2i). |
For the vector space Fn3, we consider the subset (F∗3)n. We give as follows a bijection
ψ:(F∗3)n⟶P([n])u=(u1,u2,…,un)↦ψ(u) |
where ψ(u)={i:ui=1}. Through the given map ψ, a simplicial complex Δ of P([n]) will be regarded as the simplicial complex of (F∗3)n, and be identified as a subset of Fn3 in this section without any real ambiguity.
Example 2. Let Δ be the simplicial complex of (F∗3)4 generated by {1,2} and {3,4}. Then
Δ={∅,{1},{2},{3},{4},{1,2},{3,4}} |
which is identified with
{(−1,−1,−1,−1),(1,−1,−1,−1),(−1,1,−1,−1),(−1,−1,1,−1),(−1,−1,−1,1),(1,1,−1,−1),(−1,−1,1,1)}. |
The indicator function 1Δ from Fn3 to F2 is defined by 1Δ(u)=1 only if u∈Δ. The following lemma, which is a simple consequence of the Inclusion-exclusion principle, will be used in deriving an identity involving HΔ(x1,…,xn).
Lemma 3.1. Let Δ=⟨A1,A2,…,At⟩ be a simplicial complex of (F∗3)n. Then
1Δ(u)=t∑k=1(−1)k+1∑1≤i1<i2<⋯<ik≤t1ΔAi1∩⋯∩ΔAik(u). |
Proof. Since Δ is a simplicial complex of (F∗3)n, we have Δ=∪tj=1ΔAj. The result follows from the Inclusion–exclusion principle.
Proposition 3.2. Let Δ be a simplicial complex of (F∗3)n with F the set of maximal elements of Δ. Then we have
HΔ(x1,…,xn)=1x1x2⋯xn∑∅≠S⊆F(−1)|S|+1∏i∈∩S(1+x2i) |
where we define ∏i∈∅(1+x2i)=1 by convention.
Proof. Let Δ=⟨F1,F2,…,Ft⟩, where Fi∈F. Then we see that, by Lemma 3.1,
HΔ(x1,…,xn)=∑u∈Δ1Δ(u)n∏i=1xuii=∑u∈Δt∑k=1(−1)k+1∑1≤i1<i2<⋯<ik≤t1ΔFi1∩⋯∩ΔFik(u)n∏i=1xuii=t∑k=1(−1)k+1∑1≤i1<i2<⋯<ik≤tHΔFi1∩⋯∩ΔFik(x1,…,xn)=t∑k=1(−1)k+1∑1≤i1<i2<⋯<ik≤t1x1x2⋯xn∏i∈∩kj=1Fij(1+x2i)=1x1x2⋯xn∑∅≠S⊆F(−1)|S|+1∏i∈∩S(1+x2i). |
Example 3. Let Δ be a simplicial complex of (F∗3)3 with the set of maximal element F={{1,2},{3}}. Proposition 3.2 shows that
HΔ(x1,x2,x3)=1x1x2x3(1+x21+x22+x23+x21x22)=1x1x2x3((1+x21)(1+x22)+(1+x23)−1). |
Lemma 3.3. Let Δ be a simplicial complex of (F∗3)n with F the set of maximal elements of Δ. For u∈Fn3, we have that
Re(χu(Δ))=∑∅≠S⊆F(−1)|S|+1∏i∈∩S(ζui+ζ−ui)⋅Re(∏i∉∩Sζ−ui) |
where we define ∏i∈∅(ζui+ζ−ui)=∏i∉[n]ζ−ui=1 by convention.
Proof. According to Proposition 3.2, we get that
χu(Δ)=HΔ(ζu1,…,ζun)=1ζ∑ui∑∅≠S⊆F(−1)|S|+1∏i∈∩S(1+ζ2ui)=∑∅≠S⊆F(−1)|S|+1∏i∉∩Sζ−ui∏i∈∩S(ζui+ζ−ui). |
Since ζui+ζ−ui is a real number for ui∈F3, it follows that
Re(χu(Δ))=∑∅≠S⊆F(−1)|S|+1∏i∈∩S(ζui+ζ−ui)⋅Re(∏i∉∩Sζ−ui). |
Theorem 3.4. Let Δ be a simplicial complex of (F∗3)n with one maximal element {A}. If |A|=n−1, where n≥2, there are (nm)2m codewords in the code CΔ which have the same Hamming weight
W(m):=2n−m2m−(−1)m3 |
for any integer 0≤m≤n. Moreover, the minimum distance of CΔ is W(2), which is2n−2.
Proof. If x∈F3, then
ζx+ζ−x={2,ifx=0,−1,otherwise. |
and
Re(ζ−x)={1,ifx=0,−12,otherwise. |
Since |A|=n−1, denote i0∈[n]∖A. By Lemma 3.3, for a non-zero vector u=(u1,u2,…,un) in Fn3,
Re(χu(Δ))=Re(ζ−ui0)⋅∏i∈A(ζui+ζ−ui)={(−1)n−1−k2k,ifui0=0,(−1)n−k2k−1,otherwise. |
where k=#{i:ui=0,i∈A}. According to equality (2.1), we obtain the Hamming weight of codeword cΔ(u) as follows
w(cΔ(u))={2k+12n−k−1−(−1)n−k−13,ifui0=0,2k2n−k−(−1)n−k3,otherwise. |
Let m=#{i:ui≠0,1≤i≤n}, then there are (nm)2m codewords which have the Hamming weight
w(cΔ(u))=W(m):=2n−m2m−(−1)m3. | (3.1) |
The nonzero weights W(m) in (3.1) are pairwise distinct and satisfy
W(2)<W(4)<⋯<W(2⌊n/2⌋)<W(2⌊(n−1)/2⌋+1)<W(2⌊(n−1)/2⌋−1)<⋯<W(3)<W(1). |
Hence, the minimum distance of CΔ is W(2).
Example 4. Let CΔ be a linear code defined in Theorem 3.4. If n=5, the weight distribution of the corresponding code is given in Table 1.
Weight | Frequency |
0 | 1 |
8 | 40 |
10 | 80 |
11 | 32 |
12 | 80 |
16 | 10 |
Corollary 3.5. Let Δ be a simplicial complex of (F∗3)n with one maximal element {A}. If |A|=n−1, where n≥2, then CΔ is a [2n−1,n,2n−2]-code over F3.
Proof. Since |A|=n−1, the length of CΔ is 2n−1. It then remains to prove the dimension is n. Let ei be the vector of Fn3 whose i-th coordinate is 1 and other coordinates are all zero, wi be the vector of Fn3 whose i-th coordinate is 1 and other coordinates are all −1, where 1≤i≤n. We denote by A={i1,i2,…,in−1}. Since Δ considered as a subset of Fn3 contains wi1,wi2,…,win−1, the codewords cΔ(ei) of CΔ are all nonzero. To finish the proof, we notice that cΔ(ei) are linearly independent which generate any codeword of CΔ.
For the set [n], we define
C2([n])={(A,B):A⊆[n],B⊆[n],A∩B=∅} |
to be the set of pairs of disjoint subsets of [n]. When Δ1 and Δ2 are two disjoint simplicial complexes of P([n]), we consider the set
C2(Δ1,Δ2)={(A,B):A∈Δ1,B∈Δ2}. |
Since Δ1∩Δ2=∅, we have C2(Δ1,Δ2)⊆C2([n]). Considering the vector space Fn3, there is a bijection
φ=(φ1,φ2):Fn3⟶C2([n])u=(u1,u2,…,un)↦(φ1(u),φ2(u)) |
where φ1(u)={i:ui=1} and φ2(u)={j:uj=−1}. The set C2(Δ1,Δ2) given by two disjoint simplicial complexes, under the map φ, will be then identified with the subset of Fn3 without any real ambiguity.
Example 5. Let Δ1,Δ2 be simplicial complexes of P([4]) generated by {1,2} and {3,4}. Then C2(Δ1,Δ2) consists of elements
(∅,∅)(∅,{3})(∅,{4})(∅,{3,4})({1},∅)({1},{3})({1},{4})({1},{3,4}) |
({2},∅)({2},{3})({2},{4})({2},{3,4}),({1,2},∅)({1,2},{3})({1,2}{4})({1,2},{3,4}) |
which are identified with elements of Fn3 as follows
(0,0,0,0)(0,0,−1,0)(0,0,0,−1)(0,0,−1,−1)(1,0,0,0)(1,0,−1,0)(1,0,0,−1)(1,0,−1,−1) |
(0,1,0,0)(0,1,−1,0)(0,1,0,−1)(0,1,−1,−1)(1,1,0,0)(1,1,−1,0)(1,1,0,−1)(1,1,−1,−1) |
Proposition 4.1. Let Δ1,Δ2 be simplicial complexes of P([n]) with the family of maximal elements F1 and F2 respectively. If Δ1∩Δ2=∅, then we have
HC2(Δ1,Δ2)(x1,…,xn)=∑∅≠S⊆F1∑∅≠T⊆F2(−1)|S|+|T|+2∏i∈∩S(1+xi)⋅∏j∈∩T(1+x−1j) |
where we define ∏i∈∅(1+xi)=∏j∈∅(1+x−1j)=1.
Proof.
HC2(Δ1,Δ2)(x1,…,xn)=∑(A,B)∈C2(Δ1,Δ2)∏i∈Axi∏j∈Bx−1j=(∑A∈Δ1∏i∈Axi)⋅(∑B∈Δ2∏j∈Bx−1j)=∑∅≠S⊆F1∑∅≠T⊆F2(−1)|S|+|T|+2∏i∈∩S(1+xi)⋅∏j∈∩T(1+x−1j) |
where the last equality is derived from [3,Theorem 1].
Example 6. Let Δ1,Δ2 be simplicial complexes of P([3]) with F1={{1}} and F2={{2}}. Proposition 4.1 shows that HC2(Δ1,Δ2)(x1,…,xn)=(1+x1)(1+x−12). Let u=(u1,u2,…,un), we have
Re(χu(C2(Δ1,Δ2)))=Re((1+ζu1)(1+ζ−u2))={4,ifu1=u2=0,−12,ifu1=−u2≠0,1,otherwise. |
It then follows from (2.1) that
w(cC2(Δ1,Δ2)(u))=23(|C2(Δ1,Δ2)|−Re(χu(C2(Δ1,Δ2))))={0,ifu1=u2=0,3,ifu1=−u2≠0,2,otherwise. |
It follows from (2.2) that for u∈(Fn3)∗,
w(cC2(Δ1,Δ2)c(u))={2⋅3n−1,ifu1=u2=0,2⋅3n−1−3,ifu1=−u2≠0,2⋅3n−1−2,otherwise. |
Theorem 4.2. Let Δ1=⟨{r},{s}⟩ and Δ2=⟨{t}⟩ be simplicial complexes of P([n]), where 1≤r,s,t≤n are pairwise distinct and n≥3. Then CC2(Δ1,Δ2)c is a [3n−6,n,3n−3n−1−5]-code and its weight distribution is given in Table 2.
Weight | Frequency |
0 | 1 |
Proof. The length of CC2(Δ1,Δ2)c is |C2(Δ1,Δ2)c|=3n−6 and its dimension is n according to the proof of [5,Lemma 3.6-(ⅱ)]. Since Δ1=⟨{r},{s}⟩ and Δ2=⟨{t}⟩, by Proposition 4.1, the generating function is
HC2(Δ1,Δ2)(x1,…,xn)=(1+xr)(1+x−1t)+(1+xs)(1+x−1t)−(1+x−1t)=(1+x−1t)(1+xr+xs). |
Set Bi:={(ur,us,i):ur,us∈F3∖{−i}}. Let u=(u1,u2,…,un), we have
Re(χu(C2(Δ1,Δ2)))=Re((1+ζ−ut)(1+ζur+ζus))={6,ifur=us=ut=0,3,ifur+us≠0,urus=ut=0,32,if(ur,us,ut)∈B−1∪B1,−32,ifur=us=−ut≠0,0,otherwise. |
It then follows from (2.1) that
w(cC2(Δ1,Δ2)(u))=23(|C2(Δ1,Δ2)|−Re(χu(C2(Δ1,Δ2))))={0,ifur=us=ut=0,2,ifur+us≠0,urus=ut=0,3,if(ur,us,ut)∈B−1∪B1,5,ifur=us=−ut≠0,4,otherwise. |
It follows from (2.2) that for u∈(Fn3)∗,
w(cC2(Δ1,Δ2)c(u))={3n−3n−1,ifur=us=ut=0,3n−3n−1−2,ifur+us≠0,urus=ut=0,3n−3n−1−3,if(ur,us,ut)∈B−1∪B1,3n−3n−1−5,ifur=us=−ut≠0,3n−3n−1−4,otherwise. |
The frequency of each codeword of CC2(Δ1,Δ2)c is computed by counting the vector u on its dimension.
Remark 1. Let CC2(Δ1,Δ2)c be a linear code defined in Theorem 4.2.
1). Since n≥3, then
ddmax=2⋅3n−1−52⋅3n−1>23 |
where d and dmax are the minimum and maximum weights. Hence, CC2(Δ1,Δ2)c is minimal.
2). In [5,Theorem 4.7], for instance, if p=3 and r=1, they obtain a linear code with the same parameters as CC2(Δ1,Δ2)c but with different weight distribution.
Theorem 4.3. Let Δ1=⟨{r,s}⟩ and Δ2=⟨{t}⟩ be simplicial complexes of P([n]), where 1≤r,s,t≤n are pairwise distinct and n≥3. Then CC2(Δ1,Δ2)c is an optimal [3n−8,n,3n−3n−1−6]-code and its weight distribution is given in Table 3.
Weight | Frequency |
0 | 1 |
3n−3n−1 | 3n−3−1 |
3n−3n−1−4 | 12⋅3n−3 |
3n−3n−1−5 | 6⋅3n−3 |
3n−3n−1−6 | 8⋅3n−3 |
Proof. The length of CC2(Δ1,Δ2)c is |C2(Δ1,Δ2)c|=3n−8 and its dimension is n according to the proof of [5,Lemma 3.6-(ⅱ)]. Since Δ1=⟨{r,s}⟩ and Δ2=⟨{t}⟩, by Proposition 4.1, the generating function is
HC2(Δ1,Δ2)(x1,…,xn)=(1+xr)(1+xs)(1+x−1t)=(1+x−1t)(1+xr+xs+xrxs). |
Set Mi={(ur,us,i):ur+us≠0,ur,us∈F3∖{i}}. Let u=(u1,u2,…,un), we have
Re(χu(C2(Δ1,Δ2)))=Re((1+ζ−ut)(1+ζur+ζus+ζur+us))={8,ifur=us=ut=0,12,ifur=−us≠0,ut≠0orur=us=ut≠0,−1,if(ur,us,ut)∈M−1∩M0∩M1,2,otherwise. |
It then follows from (2.1) that
w(cC2(Δ1,Δ2)(u))=23(|C2(Δ1,Δ2)|−Re(χu(C2(Δ1,Δ2))))={0,ifur=us=ut=0,5,ifur=−us≠0,ut≠0orur=us=ut≠0,6,if(ur,us,ut)∈M−1∩M0∩M1,4,otherwise. |
It follows from (2.2) that for u∈(Fn3)∗,
w(cC2(Δ1,Δ2)c(u))={3n−3n−1,ifur=us=ut=0,3n−3n−1−5,ifur=−us≠0,ut≠0orur=us=ut≠0,3n−3n−1−6,if(ur,us,ut)∈M−1∩M0∩M1,3n−3n−1−4,otherwise. |
The frequency of each codeword of CC2(Δ1,Δ2)c is computed by counting the vector u on its dimension. To check the optimality, we assume that there is a [3n−8,n,3n−3n−1−5]-code. Applying the Griesmer bound, we get that
3n−8≥n−1∑i=0⌈3n−3n−1−53i⌉=3n−7, |
which is a contradiction, so CC2(Δ1,Δ2)c is optimal.
Remark 2. Let CC2(Δ1,Δ2)c be a linear code defined in Theorem 4.3.
1). Since n≥3, then
ddmax=2⋅3n−1−32⋅3n−1>23 |
where d and dmax are the minimum and maximum weights. Hence, CC2(Δ1,Δ2)c is minimal.
2). The codes produced by our construction and the codes in [5] for p=3 have totally different parameters. Meanwhile, with a slight change of Δ1, the codes here and the codes in Theorem 4.2 are different.
The ternary codes CΔ described in Theorem 3.4 have the same parameters and weight distributions as the group character codes C3(1,n−1). Thus, the ternary codes CΔ may be viewed as the analogue of the group character codes C3(1,n−1). As a result, the codes CΔ is good for practical error detection. As pointed in [4], the weight distribution of the codes CΔ is given by the eigenvalues of the Hamming scheme. It may be interesting to investigate the relationship between these codes and the Hamming scheme.
The ternary codes CC2(Δ1,Δ2)c described in Theorem 4.2 and 4.3 have few weights and are minimal. Thus, the dual codes of CC2(Δ1,Δ2)c may be utilized to construct secret sharing schemes.
This work was supported by University Natural Science Research Project of Anhui Province under Grant No. KJ2019A0845, National Natural Science Foundation of China under Grant No.12101174 and Scientific Research Foundation of Hefei University under Grant No.18-19RC57, 18-19RC59.
All authors declare no conflicts of interest in this paper.
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1. | Zhao Hu, Yunge Xu, Nian Li, Xiangyong Zeng, Lisha Wang, Xiaohu Tang, New Constructions of Optimal Linear Codes From Simplicial Complexes, 2024, 70, 0018-9448, 1823, 10.1109/TIT.2023.3305609 |
Weight | Frequency |
0 | 1 |
8 | 40 |
10 | 80 |
11 | 32 |
12 | 80 |
16 | 10 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
3n−3n−1 | 3n−3−1 |
3n−3n−1−4 | 12⋅3n−3 |
3n−3n−1−5 | 6⋅3n−3 |
3n−3n−1−6 | 8⋅3n−3 |