# More Constructions of 3-Weight Linear Codes

Authors: Lingyong Ma; Guanjun Li; Fengyan Liu
Journal: Journal of Mathematics
 (2021)

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## Abstract

Linear codes with few weights have become an interesting research topic and important applications of cryptography and coding theory. In this paper, we apply some ternary near-bent and 2-plateaued functions orr-ary functions to construct more 3-weight linear codes, where r is a prime. Moreover, we determine the weight distributions of the resulted linear codes by means of some exponential sums.


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## Body

## 1. Introduction

Letr be a prime and q=rm. If C is a k-dimensional subspace of Frn, then C is called an n,k linear code over Fr. A code C is called an n,k,d code if d is the minimum nonzero weight in C. Let Bj be the number of code words of weight j in C. The weight enumerator of C is defined by(1)1+B1x+B2x2+⋯+Bnxn.The listBj for 1≤j≤n is called the weight distribution of C. In general, it is difficult to determine the weight distribution of a given linear code C. Information on the weight distribution of linear codes could be seen in [1–9]. If the number of nonzero Bj in 1,B1,B2,…,Bn is equal to t, then C is called a t-weight code. Linear codes with few weights have been studied due to their importance in communication and combinatorial theory [10–15]. If the parameters of C meet a bound on linear codes [16], then C is called optimal.For a subsetS=s1,s2,…,sn of Fq, we define(2)CS=ca=tras1,tras2,…,trasn:a∈Fq,where try=∑i=0s−1yri is the absolute trace function from Frs to Fr. The set S is called the defining set of CS.Many linear codes can be generated by choosing the suitable defining setS [14, 17–21]. In [22], a class of binary linear codes with 3 weights was obtained by using the exponential sum ∑y∈F2m−1tray2h+1+by. Recently, by using some r-ary bent functions, the authors obtained several classes of linear codes with 2 or 3 weights in [23–26].The Walsh transform ofgy is defined by(3)χ^gλ=∑y∈Frmζrgy−trλy,λ∈Frm,where ζr=e2π−1/r is a primitive r-th root of unity.Walsh transform is a basic tool to study the properties of cryptography functions. A functiong from Frm to Fr is bent if χ^gλ=rm/2 for all λ∈Frm. For an integer 0≤k≤m, g is called k-plateaued if χ^gλ∈0,rm+k/2 for all λ∈Frm. Plateaued function can be used as an ideal candidate function for the design of the cryptographic system because it has a variety of ideal cryptographic characteristics [27].It should be noted that binaryk-plateaued functions exist only when m and k have the same parity. The case k=0 corresponds to bent functions by definition. 1-plateaued functions are also called near-bent (see [28]), and binary 1-plateaued and 2-plateaued functions are called semibent [29].Based on the aforementioned work, this paper makes further endeavors to construct 3-weight linear codes. We use a class of quadratic ternary near-bent and 2-plateaued functions to construct two classes of ternary 3-weight linear codes. Moreover, the weight distributions of the resulted linear codes can be completely given. Numerical results indicate the proposed linear codes contain some linear codes with good parameters. By analyzing the values of some Weil sums, a class ofr-ary linear codes with 3 weights is presented.The organization of this paper is as follows. Section2 gives two classes of ternary linear codes with 3 weights. In Section 3, a class of r-ary linear codes with 3 weights is provided by using some Weil sums. Conclusions are finally drawn in Section 4.


## 2. Ternary Linear Codes with 3 Weights from Some Ternary Functions

Throughout this section, letr be an odd prime and m a positive integer. Let F3k denote the finite field with 3k elements and F3k∗=F3k,0. In [24, 26], some r-ary linear codes with 2 or 3 weights have been obtained by exploiting r-ary bent functions. We give two constructions of ternary linear codes with 3 weights by utilizing some ternary functions in this section.Letgx be a function satisfying g−x=gx and g0=0. In this section, the defining set Sa of CSa is given by(4)Sa=x∈F3m∗:gx=a,where a∈F3. Denote na=Sa. Clearly, na is the length n of CSa. We compute(5)na=13∑x∈F3m∗∑y∈F3z3ygx−a=13∑x∈F3m∗1+ζ3−aζ3gx+ζ3aζ3−gx=3m−1−1+13χ^g0+χ^g0¯ifa=0,3m−1+13ζ3−aχ^g0+ζ3aχ^g0¯ifa≠0.Note thatwtcb of cb is na−Mb, where(6)Mb=x∈F3m∗:gx=aand trbx=0,for each b∈F3m∗. Note that f−x=fx. We have(7)Mb=132∑x∈F3m∗∑y∈F3ζ3ygx−a∑z∈F3ζ3ztrbx=132∑x∈F3m∗1+ζ3−aζ3gx+ζ3aζ3−gx1+ζ3trbx+ζ3−trbx=1323m−3−3ζ3−a+ζ3a+132ζ3−aχ^g0+ζ3aχ^g0¯+2ζ3−aχ^gb+ζ3aχ^gb¯=3m−2−1+132χ^g0+χ^g0¯+2χ^gb+χ^gb¯ifa=0,3m−2+132ζ3−aχ^g0+ζ3aχ^g0¯+2ζ3−aχ^gb+ζ3aχ^gb¯ifa≠0.Generally, it is difficult to give the Walsh coefficients of a ternary function. In the following, we shall select some proper ternary plateaued functions to obtain linear codesCS0 with 3 weights over F3, where S0 is defined as in (4).Lemma 1.
(see [30], Theorem 2). Assume that k>3, λ,u,v∈F3k∗, and η is the quadratic character of F3k. Define(8)gy=trλy2+truytrvy.(i)
Iftruv/λ=2, tru2/λ=1, and trv2/λ=1, then gy is near-bent. Moreover, for a∈F3k,(9)χ^ga=ηλ−1kik+13k+1/2ζ3−tra2/λ,iftrauλtravλ=0,0,ηλ−1kik+13k+1/2ζ3−tra2/λ+1,iftrauλ,travλ=1,1or2,2,0,otherwise.(ii)
Iftruv/λ=1, tru2/λ=0, and trv2/λ=0, then gy is a 2-plateaued function. Moreover, for a∈F3m,(10)χ^ga=ηλ−1k−1ik3k/2+1ζ3−tra2/λiftrauλ,travλ=0,0,0otherwise.The following theorem gives two classes of linear codes with 3 weights overF3.Theorem 1.
Letk be a positive integer with k>3 and λ∈F3k∗. Let u,v∈F3k∗ such that trλ−1uv=2, trλ−1u2=1, and trλ−1v2=1. Let g be defined as in (8) and S0 be defined as in (4).(i)
Whenk is even, CS0 is a 3k−1−1,k code with 3 weights and has weight distribution given in Table 1(ii)
Whenk is odd, CS0 is a 3k−1+2δηλ3k−1/2−1,k code with 3 weights and has weight distribution given in Table 2, where δ=1 if k≡1mod4 and δ=−1 if k≡3mod4Table 1
Weight distribution ofCS0 for Theorem 1 (i).
WeightFrequency012⋅3k−2−2ηλ3k/2−13k−2+ηλ3k/2−12⋅3k−23k−1−2⋅3k−22⋅3k−2+2ηλ3k/2−13k−2−ηλ3k/2−1Table 2
Weight distribution ofCS0 for Theorem 1 (ii).
WeightFrequency012⋅3k−2+4δηλ3k−3/22⋅3k−12⋅3k−23k−2+2δηλ3k−3/2−12⋅3k−2+6δηλ3k−3/22⋅3k−2−2δηλ3k−3/2Proof.
Note thattrλ−1uv=2, trλ−1u2=1, and trλ−1v2=1. By Lemma 1 (i), we have χ^g0+χ^g0¯=0 as k is even. It follows from (5) that n0=3k−1−1. Hence, the length n of CS0 is n0=3k−1−1.Fora∈F3k∗, to calculate Ma, we need to calculate χ^ga+χ^ga¯ from (7).Ifk≡0mod4, then ik+1=i. We evaluate χ^ga+χ^ga¯ by distinguishing the following three cases.Whentrau/λ,trav/λ=0,0, it follows from (9) that(11)χ^ga+χ^ga¯=0if tra2λ=0,ηλ3k+1/2if tra2λ=1,−ηλ3k+1/2if tra2λ=2.Whentrau/λ,trav/λ=1,1 or 2,2, it follows from (9) that(12)χ^ga+χ^ga¯=−ηλ3k+1/2if tra2λ=0,ηλ3k+1/2if tra2λ=1,0if tra2λ=2.Whentrau/λ,trav/λ∈F32,0,0,1,1,2,2,(13)χ^ga+χ^ga¯=0.It then follows from (7) with b=0 that(14)Ma∈3k−2−1,3k−2−2ηλ3k−2/2−1,3k−2+2ηλ3k−2/2−1.Hence, the Hamming weightwtca of ca satisfies(15)wtca=n0−Ma∈2⋅3k−2−2ηλ3k−2/2,2⋅3k−2,2⋅3k−2+2ηλ3k−2/2.This implies that the codeCS0 has 3 weights in the above set.Ifk≡2mod4, then ik+1=−i. Similar to the proof above, we obtain(16)wtca=n0−Ma∈2⋅3k−2−2ηλ3k−2/2,2⋅3k−2,2⋅3k−2+2ηλ3k−2/2,for eacha∈F3k∗. Since g−x=0 if gx=0 and 0∉S0, we have the minimum distance of CS0⊥ is 2. It is easy to see that the number B2⊥ of code words with weight 2 in CS0⊥ is n. The first three Pless power moments give the desired weight distribution. Since k>3, wtca>0 for any a∈F3k∗. Hence, the dimension of CS0 is k.(ii)
Conclusion (ii) can be proved by the similar proof as (i).Example 1.
Letk=4, λ=1, u=−1, and v=1. Define gy=try2−try2. Then, CS0 in Theorem 1 (i) is a ternary 26,4,12 code, and its weight enumerator is 1+12z12+62z18+6z24.Theorem 2.
Letk be a positive integer with k>4 and λ∈F3k∗. Let u,v∈F3k∗ such that trλ−1uv=1, trλ−1u2=0, and trλ−1v2=0. Let S0 be defined as in (4) and f be defined as in (8).(i)
Whenk is even, CS0 is a 3k−1−2δηλ3k/2−1,k 3-weight code and has weight distribution listed in Table 3, where δ=1 if k≡0mod4 and δ=−1 if k≡2mod4(ii)
Whenk is odd, CS0 is a 3k−1−1,k 3-weight code and has weight distribution listed in Table 4Table 3
Weight distribution ofCS0 for Theorem 2 (i).
WeightFrequency012⋅3k−2−4δηλ3k/2−13k−3k−22⋅3k−23k−3−2δηλ3k/2−2−12⋅3k−2−6δηλ3k/2−12⋅3k−3+2δηλ3k/2−2Table 4
Weight distribution ofCS0 for Theorem 2 (ii).
WeightFrequency012⋅3k−2−2ηλ3k+1/2−13k−3+ηλ3k−3/22⋅3k−23k−2⋅3k−3−12⋅3k−2+2ηλ3k+1/2−13k−3−ηλ3k−3/2Proof.
We just omit the proof here since it is similar to that of Theorem1.Remark 1.
It should be noted that ifλ is a nonsquare in F34∗, the first conclusion in Theorem 2 is also true when k=4.Example 2.
Assumek=4 and β is the generator of F34∗, where β is a root of x4−x3−1=0. Let λ=β, u=β16, and v=β8. Define gx=trβx2+trβ16xtrβ8x. Then, the code CS0 in Theorem 2 (i) is 44,4,18 code and has weight enumerator 1+4z18+72z30+4z36.
Sincegx=g−x, the set S0 defined by (4) can be expressed as(17)S0=−S¯∪S¯,where didj−1≠±1 for di,dj∈S¯ and di≠dj. Selecting D¯ as the defining set can give more ternary 3-weight codes.Corollary 1.
Assume thatk>3 and λ∈F3k∗. Let u,v∈F3k∗ such that trλ−1uv=2, trλ−1u2=1, and trλ−1v2=1. Let D¯ be given as in (17) and g be given as in (8).(i)
Whenk is even, CS¯ is a 3k−1−1/2,k 3-weight code and has weight distribution given in Table 5(ii)
Whenk is odd, CS¯ is a 3k−1−1/2+δηλ3k−1/2,k 3-weight code and has weight distribution given in Table 6, where δ=1 if k≡1mod4 and δ=−1 if k≡3mod4Table 5
Weight distribution ofCS¯ for Corollary 1 (i).
WeightFrequency013k−2−ηλ3k/2−13k−2+ηλ3k/2−13k−23k−1−2⋅3k−23k−2+ηλ3k/2−13k−2−ηλ3k/2−1Table 6
Weight distribution ofCS¯ for Corollary 1 (ii).
WeightFrequency013k−2+2δηλ3k−3/22⋅3k−13k−23k−2+2δηλ3k−3/2−13k−2+3δηλ3k−3/22⋅3k−2−2δηλ3k−3/2Example 3.
Letk=4. Let λ=1, u=−1, and v=1. Define gy=try2−try2. The code CS¯ in Corollary 1 (i) is a 13,4,6 code and has weight enumerator 1+12z6+62z9+6z12.Example 4.
Letk=5. Let λ=−1, u=−1, and v=1. Define gy=tr−y2−try2. Then, CS¯ in Corollary 1 (ii) is a 31,5,18 code and has weight enumerator 1+60z18+162z21+20z27. By using the Griesmer bound, we know this code is optimal.Corollary 2.
Assume thatk>4 and λ∈F3k∗. Let u,v∈F3k∗ such that trλ−1uv=1, trλ−1u2=0, and trλ−1v2=0. Let S¯ be defined as in (17) and f be defined as in (8).(i)
Whenk is even, CS¯ is a 3k−1−1/2−δηλ3k/2,k 3-weight code, and its weight distribution is listed in Table 7, where δ=1 if k≡0mod4 and δ=−1 if k≡2mod4(ii)
Whenk is odd, CS¯ is a 3k−1−1/2,k 3-weight code and has weight distribution listed in Table 8Table 7
Weight distribution ofCS¯ for Corollary 2 (i).
WeightFrequency013k−2−2δηλ3k/2−13k−3k−23k−23k−3−2δηλ3k/2−2−13k−2−3δηλ3k/2−12⋅3k−3+2δηλ3k/2−2Table 8
Weight distribution ofCS¯ for Corollary 2 (ii).
WeightFrequency013k−2−ηλ3k+1/2−13k−3+ηλ3k−3/23k−23m−2⋅3m−3−13k−2+ηλ3k+1/2−13k−3−ηλ3k−3/2Example 5.
Assumeβ is a generator of F34∗, where β is a root of x4−x3−1=0. Let λ=β, u=β16, and v=β8. Define gy=trβy2+trβ16ytrβ8y. Then, the code CS¯ in Corollary 2 (i) is a 22,4,9 code and has weight enumerator 1+4z9+72z15+4z18.


## 3. A Class ofr-Ary Linear Codes with 3 Weights

Denoted=gcdh,m, where m and h are positive integers. For λ,a∈Frm, let fx=trλxrh+1.Based on the fact that a quadraticr-ary bent function has full rank, Zhou et al. ([26], Section III.B) obtained a class of linear codes with 2 or 3 weights by using Gold functions. Following the work of Zhou et al. [26], we shall employ Gold class of plateaued functions whose rank is less than m to construct r-ary linear codes with 3 weights.Forλ,a∈Frm, we denote by Whλ,a the Weil sum given by(18)Whλ,a=∑x∈Frmζrtrλxrh+1+ax.Lemma 2.
(see [31], Theorem 4.1). For m=2k, the equation λrhxr2h+λx=0 has solutions in Frm∗ if and only if m/d is even and λrm−1/rd+1=−1k/d.
In [31, 32], the following two lemmas studied explicit evaluations of the exponential sum (18).Lemma 3.
(see [31]). Let m/d be even with m=2k. Then, Whλ,0=Wh−λ,0, and(19)Whλ,0=rk+difλrm−1/rd+1=−1k/dandkdis odd,−rk+difλrm−1/rd+1=−1k/dandkdis even.Lemma 4.
(see [32]). Let m/d be even with m=2k. Assume that gx=λrhxr2h+λx is not a permutation polynomial over Frm. Then, for a≠0, we have Whλ,a=0 unless the equation gx=−arh is solvable. If this equation is solvable, then(20)Whλ,a=−−1k/drk+dζr−trλx0rh+1,where x0 is such that gx0=−arh.Letm/d be even with m=2k and λ∈Frm∗ such that λrm−1/rd+1=−1k/d. It follows from Lemmas 3 and 4 and the definition of the plateaued function that fx=trλxrh+1 is a 2d-plateaued function if λrm−1/rd+1=−1k/d.Lemma 5.
Letm/d be even with m=2k and λ∈Frm∗ such that λrm−1/rd+1=−1k/d. Then,(21)Whλ,0=Wh2λ,0=⋯=Whr−1λ,0=rk+difλrm−1/rd+1=−1k/dandkdis odd,−rk+difλrm−1/rd+1=−1k/dandkdis even.Proof.
Sincem/d is even, r−1|rm−1/rd+1. Hence, cλrm−1/rd+1=λrm−1/rd+1=−1k/d for each c∈Fr∗. The desired result follows from Lemma 3.Lemma 6.
Letm/d be even with m=2k and λ∈Frm∗ such that λrm−1/rd+1=−1k/d. For a∈Frm∗, we have(22)∑z∈Fr∗∑y∈Fr∗∑x∈Frmζrtryλxrh+1+zax∈0,−−1k/dr−12rk+d,−1k/drk+dr−1.Proof.
Sincem/d is even and λrm−1/rd+1=−1k/d, the equation λrhxr2h+λx=0 is solvable for x∈Frm∗ by Lemma 2. For a∈Frm∗, we claim that the equation λrhxr2h+λx+arh=0 is solvable if and only if the equation yλrhxr2h+yλx+zarh=0 is solvable for each y,z∈Fr∗×Fr∗. Suppose x0 is a solution of λrhxr2h+λx+arh=0. Clearly, x0 is also a root of y−1zλrhxr2h+y−1zλx+y−1zarh=0. Then, y−1zλrhx0r2h+y−1zλx0+y−1zarh=0. We can rewrite this as λrhy−1zx0r2h+λy−1zx0+y−1zarh=0 since z,y∈Fr∗. It then follows that yλrhy−1zx0r2h+yλy−1zx0+zarh=0. That is to say, the equation yλrhxr2h+yλx+zarh=0 has a solution y−1zx0. Conversely, assume x0 is a solution of yλrhxr2h+yλx+zarh=0; then, yλrhx0r2h+yλx0+zarh=0. Note that y,z∈Fr∗. Hence, we have λrhz−1yx0r2h+λz−1yx0+arh=0. This implies that λrhxr2h+λx+arh=0 has a solution z−1yx0.
Ifλrhxr2h+λx+arh=0 has no solutions in Frm, then by Lemma 4 and the discussions above, we have(23)∑y∈Fr∗∑z∈Fr∗∑x∈Frmζrtryλxrh+1+zax=0.
Ifλrhxr2h+λx+arh=0 has a solution x0 in Frm, then we know that yλrhxr2h+yλx+zarh=0 has a solution y−1zx0 from the discussions above. It follows from (20) that(24)∑z∈Fr∗∑y∈Fr∗∑x∈Frmζrtryλxrh+1+zax=−1k/d+1rk+d∑z∈Fr∗∑y∈Fr∗ζr−tryλy−1zx0rh+1=−1k/d+1rk+d∑z∈Fr∗∑y∈Fr∗ζr−y−1z2trλx0rh+1=−1k/drk+dr−1if trλx0rh+1≠0,−1k/d+1rk+dr−12if trλx0rh+1=0.
Hence, for anya∈Frm∗,(25)∑z∈Fr∗∑y∈Fr∗∑x∈Frmζrtryλxrh+1+zax∈0,−1k/drk+dr−1,−1k/d+1rk+dr−12.
Leth be a positive integer, 1≤h<k, where m=2k>4 is an even integer. Define(26)S=x∈Frm∗:trλxrh+1=0,where λ∈Frm∗.Theorem 3.
Leth be a positive integer, 1≤h<k, where m=2k>4 is an even integer. Let S be defined as in (26), where λ∈Frm∗. Define d=gcdh,m. Assume that m/d is even and λrm−1/rd+1=−1k/d.(i)
Ifk/d is odd, then CS is a r−1rk+d−1+rm−1−1,m,rm−2r−1 3-weight code and has weight distribution given in Table 9(ii)
Ifk/d is even, then CS is a r−1rk+d−1−rm−1−1,m,rm−2r−1−r−1rk+d−1 3-weight code with the weight distribution given in Table 10Table 9
Weight distribution ofCS for Theorem 3 (i).
WeightFrequency01rm−2r−1rm−2−rk+d−2rm−2+rk+d−1/rm+2d−3rm−2r−1+r−12rk+d−2rm−1−rm−2−rk+d−2rm−1+rk+d−1/rm+2d−3rm−2r−1+r−1rk+d−1rm−2−rk+d−2r−1rm−2/rm+2d−3Table 10
Weight distribution ofCS for Theorem 3 (ii).
WeightFrequency01rm−2r−1−r−1rk+d−1rm−2+rk+d−2r−1rm−2/rm+2d−3rm−2r−1−r−12rk+d−2rm−1−rm−2+rk+d−2rm−1−rk+d−1/rm+2d−3rm−2r−1rm−2+rk+d−2rm−2−rk+d−1/rm+2d−3Proof.
We only present the proof of (i) since the others can be proven by a similar proof. DenotenS=S∪0. We have(27)nS=1r∑x∈Frm∑y∈Frζrytrλxrh+1=1r∑x∈Frm1+ζrtrλxrh+1+ζr2trλxrh+1+⋯+ζrr−1trλxrh+1=rm−1+1rWhλ,0+Wh2λ,0+⋯+Whr−1λ,0.
Note thatk/d is odd and λrm−1/rd+1=−1k/d. By Lemma 5, we have(28)n=nS−1=rm−1+r−1rk+d−1−1.
For eacha∈Frm∗, let(29)Ma=x∈Frm:trλxrh+1=0and trax=0.
Then, the Hamming weightwtca is nS−Ma. We can compute(30)Ma=1r2∑x∈Frm∑y∈Frζrytrλxrh+1∑z∈Frζrztrax=rm−2+1r2∑y∈Fr∗∑x∈Frmζrytrλxrh+1+1r2∑y∈Fr∗∑z∈Fr∗∑x∈Frmζrtryλxrh+1+zax=rm−2+1r2∑y∈Fr∗Whyλ,0+1r2∑y∈Fr∗∑z∈Fr∗∑x∈Frmζrtryλxrh+1+zax.
It follows from Lemmas5 and 6 that(31)Ma∈rm−2+r−12rk+d−2,rm−2,rm−2+r−1rk+d−2.
For eacha∈Frm∗, we have(32)wtca=rm−2r−1,r−1rm−2+r−1rk+d−1,rm−2r−1+r−12rk+d−2.
Then,CS has three nonzero weights above. When m>4, for each a∈F3m∗, wtca>0, and thus, the dimension of the code is m.
Note that0∉S. The minimum distance of CS⊥ cannot be one. Since trλcxrh+1=0 for all c∈Fr∗, if trλxrh+1=0, the minimum Hamming weight of CS⊥ is 2. Moreover, we can prove that B2⊥ is equal to nr−12. In fact, for each r−1 distinct element v,2v,…,r−1v∈S, we can get r−1r−12 code words with weight 2 in CS⊥. Define w1=rm−2r−1, w2=rm−2r−1+rk+d−2r−12, and w3=rm−2r−1+rr−1rk+d−2.
Solving the following linear equations gives the desired weight distribution:(33)Bw1+Bw2+Bw3=rm−1,w1Bw1+w2Bw2+w3Bw3=r−1nrm−1,w12Bw1+w22Bw2+w32Bw3=r−1nr−1n+1+2r−12nrm−2.Example 6.
Letm=8 and h=2. Let fx=trx10. Then, the code CS is a 1700,8,972 code and has weight enumerator 1+60z972+6480z1134+20z1458.Example 7.
Letm=6 and h=1. Let β be the generator of F56∗, where β is a root of x6+x4−x3+x2+2=0. Let λ=β3. Clearly, λ56−1/6=β7812=−1. Let fx=trβ3x6. Then, the code CS is a 3624,6,2500 code and has weight enumerator 1+144z2500+15000z2900+480z3000.


## 4. Conclusion

We succeed in constructing some linear codes with 3 weights from somer-ary functions with few Walsh transform values in the odd characteristic. The weight distributions of linear codes in this paper are determined by analyzing Walsh coefficients of r-ary functions. Compared with other linear codes with 3 weights, the parameters of the linear codes with 3 weights of this paper are different from those obtained from bent functions in [19, 23, 24, 26]. As applications, some of the 3-weight codes can be used to obtain secret sharing schemes [15] and association schemes introduced in [11].



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