Two new families of two-weight codes
aa r X i v : . [ c s . I T ] S e p JOURNAL OF L A TEX CLASS FILES, VOL. , NO. , 2017 1
Two new families of two-weight codes ∗ Minjia Shi, Yue Guan, and Patrick Sol´e
Abstract —We construct two new infinite families oftrace codes of dimension m , over the ring F p + u F p , with u = u, when p is an odd prime. They have thealgebraic structure of abelian codes. Their Lee weightdistribution is computed by using Gauss sums. By Graymapping, we obtain two infinite families of linear p -arycodes of respective lengths ( p m − and p m − . When m is singly-even, the first family gives five-weightcodes. When m is odd, and p ≡ , the firstfamily yields p -ary two-weight codes, which are shownto be optimal by application of the Griesmer bound. Thesecond family consists of two-weight codes that are shownto be optimal, by the Griesmer bound, whenever p = 3 and m ≥ , or p ≥ and m ≥ . Applications to secretsharing schemes are given.
Index Terms —Two-weight codes; Gauss sums; Gries-mer bound; Secret sharing schemes.
I. I
NTRODUCTION T Wo-weight codes over fields have been stud-ied since the 1970s due to their connections tostrongly regular graphs, finite geometries and differ-ence sets [5]. However, most constructions, have usedcyclic codes over finite fields [3], [4]. In the presentpaper, we use trace codes over a semi-local ring whichis a quadratic extension of a finite field, and obtaincodes over a finite field by Gray mapping. Trace codesare naturally low-rate codes, and not necessarily cyclic.This is part of a general research program where avariety of few weight codes are obtained by varying thealphabet ring and the defining set [14], [15], [16]. Herewe consider an alphabet ring of odd characteristic, incontrast with [14], [15], and over a non local ring, incontrast with [16]. We consider two families depending
MinJia Shi is with Key Laboratory of Intelligent Computing &Signal Processing, Ministry of Education, Anhui University No. 3Feixi Road, Hefei Anhui Province 230039, P. R. China, NationalMobile Communications Research Laboratory, Southeast Univer-sity, 210096, Nanjing, P. R. China and School of MathematicalSciences of Anhui University, Anhui, 230601, P. R. China (E-mail:[email protected]).Yue Guan is with School of Mathematical Sciences of AnhuiUniversity, Anhui, 230601, P. R. China.Patrick Sol´e is with CNRS/ LAGA, University Paris 8, 93 526Saint-Denis, FranceManuscript received November 11, 2016; revised May 14, 2017.M. Shi was supported in part by the National Natural ScienceFoundation of China under Grant 61672036, in part by the Tech-nology Foundation for Selected Overseas Chinese Scholar, Ministryof Personnel, China, under Grant 05015133, in part by the OpenResearch Fund of National Mobile Communications Research Lab-oratory, Southeast University, under Grant 2015D11, and in part bythe Key Projects of Support Program for Outstanding Young Talentsin Colleges and Universities under Grant gxyqZD2016008. on two different defining sets. These codes are visiblyabelian, but possibly not cyclic. The field image of thefirst family has two or five weights, depending on thechoice of parameters. The second family only containstwo-weight codes. Note that abelian codes over ringshave been studied already in [11]. The alphabet ringwe consider here is F p + u F p , with u = u. It is a semi-local ring, which is ring isomorphic to F p × F p (Cf. § F r + v F r , with v = v. The latter ring has beenemployed recently to construct convolutional codesover fields [9]. Some bounds on codes over F p + u F p , with u = u can be found in [10].The defining set of our abelian code is not a cyclicgroup, but it is an abelian group. The defining set ofthe first family is related to quadratic residues in anextension of degree m of F p , which makes quadraticGauss sums appear naturally in the weight distributionanalysis, and requires p to be an odd prime. When m isodd, and p ≡ , we obtain an infinite familyof linear p -ary two-weight codes, which are shown tobe optimal by application of the Griesmer bound. Thecodes in the second family are also shown to be optimalby the same technique up to finitely many exceptions.We show that, both in the five-weight and in the two-weight cases, the first family has a very nice supportinclusion structure which makes it suitable for use in aMassey secret sharing scheme [7], [8], [19]. Indeed, wecan show that all nonzero codewords are minimal forthe poset of codewords ordered by support inclusion. Asimilar result holds for the second family. To the bestof our knowledge, the weight distributions of the codesobtained here are different from the classical familiesof [4] and from the those of the codes in [14], [15],[16]. Our codes are therefore new.The paper is organized as follows. Section II collectsthe notions and notations needed in the rest of thearticle. Section III shows that the trace codes areabelian. Section IV recalls and reproves some resultson Gaussian periods. Section V computes the weightdistribution of our codes, building on the charactersum evaluation of the preceding section. Section VIdiscusses the optimality of the p -image from tracecodes over R . Section VII determines the minimumdistance of the dual codes. Section VIII determinesthe support structure of the p -ary image and describesan application to secret sharing schemes. OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 2017 2
II. D
EFINITIONS AND NOTATIONS
A. Rings
Consider the ring R = F p + u F p where u = u and p is a odd prime. It is semi-local with maximal ideals ( u ) and ( u − . For any integer m ≥ , we construct anextension of degree m as R = F p m + u F p m with again u = u. There is a Frobenius operator F which maps a + ub onto a p + ub p . The
Trace function, denotedby
T r , is defined as
T r = m − P j =0 F j . It follows fromthese definitions that
T r ( a + ub ) = tr ( a ) + utr ( b ) , for a, b ∈ F p m . Here tr () denotes the absolute trace of F p m , given by tr ( z ) = z + z p + · · · + z p m − , z ∈ F p m . The ring R is semi-local with maximal ideals ( u ) and ( u − , and respective quotients R / ( u ) and R / ( u − are both isomorphic to F p m . The ChineseRemainder Theorem shows that u F p m + (1 − u ) F p m isisomorphic to the product ring F p m × F p m . Similarly,the group of units R ∗ is u F ∗ p m + (1 − u ) F ∗ p m whichis isomorphic to F ∗ p m × F ∗ p m . Here F ∗ p m denotes themultiplicative group of F p m . Denote the squares andthe non-squares of F p m by Q and N , respectively.Thus Q = { x | x ∈ F ∗ p m } , N = F ∗ p m \ Q . We write L = u Q + (1 − u ) F ∗ p m and let L ′ = R ∗ for simplicity. Thus L is a subgroup of R ∗ of index . B. Gray map
As a preparation for the image code from trace codesover R , we shall take a closer look at the Gray map φ from R to F p , which is defined by φ ( a + ub ) =( − b, a + b ) for a, b ∈ F p . It is a one to one map from R to F p , which extends naturally into a map from R n to F np . Denote the Hamming weight on F np by w H ( . ) , and the Hamming distance on F np by d H ( ., . ) . The
Lee weight is defined as the Hamming weight of theGray image w L ( a + ub ) = w H ( − b ) + w H (2 a + b ) for a, b ∈ F np . The
Lee distance of x, y ∈ R n is defined as d L ( x, y ) = w L ( x − y ) . Thus the Gray map is a linearisometry from ( R n , d L ) to ( F np , d H ) . For convenience,we write N = 2 n in the rest of the paper. C. Codes A linear code C over R of length n is an R -submodule of R n . If x = ( x , x , · · · , x n ) and y = ( y , y , · · · , y n ) are two elements of R n , theirstandard inner product is defined by h x, y i = n P i =1 x i y i ,where the operation is performed in R . The dualcode of C is denoted by C ⊥ and defined as C ⊥ = { y ∈ R n |h x, y i = 0 , ∀ x ∈ C } . By definition, C ⊥ is also a linear code over R . Given a finite abeliangroup G, a code over R is said to be abelian [2],if it is an ideal of the group ring R [ G ] . Recall thatthe ring R [ G ] is defined on functions from G to R with pointwise addition as addition, and convolutionproduct as multiplication. Concretely, it is the set ofall formal sums f = P h ∈ G f h X h , with addition andmultiplication defined as follows. If f, g ∈ R [ G ] , wewrite f + g = X g ∈ G ( f h + g h ) X h , and f g = X h ∈ G ( X r + s = h f r g s ) X h . In other words, the coordinates of C are indexed byelements of G and G acts regularly on this set. Formore details on abelian codes see [18]. In the specialcase when G is cyclic, the code is a cyclic code in theusual sense [12]. III. S YMMETRY
For a ∈ R , define the vector ev ( a ) by the followingevaluation map ev ( a ) = ( T r ( ax )) x ∈ L . Define the code C ( m, p ) by the formula C ( m, p ) = { ev ( a ) | a ∈ R} .Thus C ( m, p ) is a code of length | L | = ( p m − andsize | R | m over R. Similarly, define the vector ev ′ ( a ) bythe following evaluation map ev ′ ( a ) = ( T r ( ax )) x ∈ L ′ , and the code C ′ ( m, p ) by the formula C ′ ( m, p ) = { ev ′ ( a ) | a ∈ R} . Thus C ′ ( m, p ) is a code of length | L ′ | = ( p m − and size | R | m over R. Proposition 3.1
The group L ( resp. L ′ ) acts reg-ularly on the coordinates of C ( m, p ) ( resp. C ′ ( m, p )) . Proof
For any w, v ∈ L the change of variables x ( v/w ) x maps w to v. This transformation defines thusa transitive action of L on itself. Given an ordered pair ( w, v ) this transformation is unique, hence the actionis regular. A similar argument holds for C ′ ( m, p ) and L ′ . The code C ( m, p ) is thus an abelian code withrespect to the group R ∗ . In other words, it is an idealof the group ring R [ R ∗ ] . As observed in the previoussection R ∗ is not a cyclic group, hence C ( m, p ) maybe not cyclic. IV. C HARACTER SUMS
In this section we give some background material oncharacter sums. Let χ denote an arbitrary multiplicativecharacter of F q . Assume q is odd. Denoted by η the quadratic multiplicative character is defined by η ( x ) = 1 , if x is a square and η ( x ) = − , if not.Let ψ denote the standard canonical additive characterof F q . The squares and the non-squares of F q are OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 2017 3 denoted, extending the notation of § Q and N , respectively. Thus, Q = { x | x ∈ F ∗ q } , N = F ∗ q \ Q . The classical
Gauss sum can be defined as G ( χ ) = P x ∈ F ∗ q ψ ( x ) χ ( x ) . We define the following character sums Q = X x ∈Q ψ ( x ) , N = X x ∈N ψ ( x ) . On the basis of orthogonality of characters [12, Lemma9, p. 143] it is evident that Q + N = − . Noting thatthe characteristic function of Q is η , we get then Q = G ( η ) − , N = − G ( η ) − . It is well known [6] that if q = p m , the quadraticGauss sums can be evaluated as G ( η ) = ( − m − √ q , p ≡ , (1) G ( η ) = ( − m − i m √ q, p ≡ . (2)Particularly, if m is singly-even, these formulascan be simplified to G ( η ) = ǫ ( p ) √ q, with ǫ ( p ) =( − ( p +1)2 , yielding Q = ǫ ( p ) √ q − , N = − ǫ ( p ) √ q + 12 . In fact Q and N are examples of Gaussian periods ,and these relations could have been deduced from [6,Lemma 11].V. W
EIGHT DISTRIBUTIONS OF TRACE CODES
Let ω = exp( πip ) be a complex root of unity oforder p. If y = ( y , y , · · · , y N ) ∈ F Np , let Θ( y ) = N P j =1 ω y j . For simplicity, we let θ ( a ) = Θ( φ ( ev ( a ))) . By linearity of the Gray map, and of the evaluationmap, we see that θ ( sa ) = Θ( φ ( ev ( sa ))) , for any s ∈ F ∗ p . For our purpose, let us begin with the followingcorrelation lemma.
Lemma 5.1 [15] For all y = ( y , y , · · · , y N ) ∈ F Np , we have p − P s =1 Θ( sy ) = ( p − N − pw H ( y ) . In connection with the proceding discussion, we nowdistinguish two cases of weight distributions dependingon the defining set.
A. The case of L = Q × F ∗ p m m is singly-even: Theorem 5.2
Assume m issingly-even. For a ∈ R , the Lee weight of codewordsof C ( m, p ) is given below.(a) If a = 0 , then w L ( ev ( a )) = 0 ;(b) If a = uα , α ∈ F ∗ p m , then if α ∈ Q then w L ( ev ( a )) = ( p − (cid:0) p m − − p m − − ǫ ( p ) p m/ − + ǫ ( p ) p m/ − (cid:1) , α ∈ N then w L ( ev ( a )) = ( p − (cid:0) p m − − p m − + ǫ ( p ) p m/ − − ǫ ( p ) p m/ − (cid:1) ; (c) If a = (1 − u ) β , β ∈ F ∗ p m , then w L ( ev ( a )) =( p − p m − − p m − ) ;(d) If a = uα + (1 − u ) β ∈ R ∗ , then if α ∈ Q then w L ( ev ( a )) = ( p − (cid:0) p m − − p m − + ǫ ( p ) p m/ − (cid:1) , α ∈ N then w L ( ev ( a )) = ( p − (cid:0) p m − − p m − − ǫ ( p ) p m/ − (cid:1) . Proof (a) If a = 0 , then T r ( ax ) = 0 . So w L ( ev ( a )) = 0 .(b) If a = uα, x = ut + (1 − u ) t ′ with α ∈ F ∗ p m , then ax = uαt, and T r ( ax ) = T r ( uαt ) = utr ( αt ) . Taking Gray map yields φ ( ev ( a )) =( − tr ( αt ) , tr ( αt )) t,t ′ . Taking character sums θ ( a ) = X t ∈Q X t ′ ∈ F ∗ pm ω − tr ( αt ) + X t ∈Q X t ′ ∈ F ∗ pm ω tr ( αt ) = 2 X t ∈Q X t ′ ∈ F ∗ pm ω − tr ( αt ) = 2( p m − X t ∈Q ω tr ( αt ) . Replaced αt by t , it is easy to check that the lastcharacter sum is Q or N depending on α ∈ Q or α ∈ N . Since m is even, s ∈ F ∗ p is asquare in F p m . Thus θ ( sa ) = θ ( a ) , for any s ∈ F ∗ p . The statement follows from Lemma 5.1.Thus w L ( ev ( a )) = p − p ( N − p m − Q ) , or w L ( ev ( a )) = p − p ( N − p m − N ) , accordingto the value of η ( α ) . (c) If a = (1 − u ) β, x = ut + (1 − u ) t ′ with β ∈ F ∗ p m , then ax = βt ′ − uβt ′ , and T r ( ax ) = tr ( βt ′ ) − utr ( βt ′ ) . Taking Gray map yields φ ( ev ( a )) = ( tr ( βt ′ ) , tr ( βt ′ )) t,t ′ . Taking charac-ter sums θ ( a ) = 2 P t ∈Q P t ′ ∈ F ∗ pm ω tr ( βt ′ ) = 1 − p m . Thus w L ( ev ( a )) = ( p − p m − − p m − ) . (d) Let a = uα + (1 − u ) β ∈ R ∗ , x = ut +(1 − u ) t ′ . So T r ( ax ) = tr ( βt ′ ) + utr ( αt − βt ′ ) . Thus φ ( ev ( a )) = ( − tr ( αt − βt ′ ) , tr ( αt + βt ′ )) t,t ′ by the Gray map. Taking charactersums θ ( a ) = P t ∈Q ω − tr ( αt ) P t ′ ∈ F ∗ pm ω tr ( βt ′ ) + P t ∈Q ω tr ( αt ) P t ′ ∈ F ∗ pm ω tr ( βt ′ ) = − P t ∈Q ω tr ( αt ) . Bya change of variable t = αt, we see that the lastcharacter sum is Q or N depending on α ∈ Q or α ∈ N . Thus w L ( ev ( a )) = p − p ( N + 2 Q ) , or w L ( ev ( a )) = p − p ( N + 2 N ) , considering thevalue of η ( α ) . Therefore, we have constructed a p -ary code oflength N = ( p m − , dimension m, with five OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 2017 4 weights. The weight distribution is given in Table I.
Table I . weight distribution of C ( m, p ) inTheorem 5.2Weight Frequency0 1 ( p − p m − − p m/ − )( p m − p m − ( p − p m − − p m − − p m/ − ) ( p m − ( p − p m − − p m − + p m/ − ) ( p m − ( p − p m − − p m − ) p m − p − p m − + p m/ − )( p m − p m − (Note that taking ǫ ( p ) = 1 , or − , leads to the samevalues.) m is odd and p ≡ : Note that in thatcase by (2) in Section IV we see that G ( η ) is imag-inary. This implies that ℜ ( Q ) = ℜ ( N ) = − , where ℜ ( z ) denotes the real part of the complex number z. We need first to refine the following correlation lemma.
Lemma 5.3 [15] If p ≡ , then we have p − P s =1 θ ( sa ) = ( p − ℜ ( θ ( a )) . Theorem 5.4
Assume m is odd and p ≡ . For a ∈ R , the Lee weight of codewords of C ( m, p ) is given below.(a) If a = 0 , then w L ( ev ( a )) = 0 ;(b) If a = uβ with β ∈ F ∗ p m , then w L ( ev ( a )) =( p − p m − − p m − ) ;(c) If a = (1 − u ) β with β ∈ F ∗ p m , then w L ( ev ( a )) =( p − p m − − p m − ) ;(d) If a ∈ R ∗ , then w L ( ev ( a )) = ( p − p m − − p m − ) . Proof
The proof of the case (a) is like that ofTheorem 5.2. The case (b) follows from Lemma 5.3applied to the correlation lemma. Thus ℜ ( θ ( a )) =1 − p m , and pw L ( ev ( a )) = ( p − N − ℜ ( θ ( a ))) , yielding w L ( ev ( a )) = ( p − p m − − p m − ) . Theresult follows. The proof of case (c) is the same asthat of case (b). In the case (d), ℜ ( θ ( a )) = 1 , then w L ( ev ( a )) = ( p − p m − − p m − ) . Thus we obtain a family of p -ary two-weight codesof parameters [ p m − p m + 1 , m ] , with weightdistribution as given in Table II. The parameters aredifferent from those in [4], [14], [15] and [16]. Table II . weight distribution of C ( m, p ) inTheorem 5.4Weight Frequency0 1 ( p − p m − − p m − ) ( p m − ( p − p m − − p m − ) 2( p m − B. The case of L ′ = F ∗ p m × F ∗ p m Theorem 5.5
For a ∈ R , the Lee weight ofcodewords of C ′ ( m, p ) is (a) If a = 0 , then w L ( ev ′ ( a )) = 0; (b) If a = uα , α ∈ F ∗ p m , then w L ( ev ′ ( a )) = 2( p − p m − − p m − ); (c) If a = (1 − u ) β , β ∈ F ∗ p m , then w L ( ev ′ ( a )) =2( p − p m − − p m − ); (d) If a ∈ R ∗ , then w L ( ev ′ ( a )) = 2( p − p m − − p m − ) . Proof (a) If a = 0 , then T r ( ax ) = 0 . So w L ( ev ′ ( a )) = 0 .(b) If a = uα, x = ut + (1 − u ) t ′ with α ∈ F ∗ p m , then ax = uαt, and T r ( ax ) = T r ( uαt ) = utr ( αt ) . Taking Gray map yields φ ( ev ′ ( a )) = ( − tr ( αt ) , tr ( αt )) t,t ′ . Taking char-acter sums θ ( a ) = P t ∈ F ∗ pm P t ′ ∈ F ∗ pm ω − tr ( αt ) + P t ∈ F ∗ pm P t ′ ∈ F ∗ pm ω tr ( αt ) = 2 P t ∈ F ∗ pm P t ′ ∈ F ∗ pm ω − tr ( αt ) = − p m − . Thus w L ( ev ′ ( a )) = 2( p − p m − − p m − ) . (c) If a = (1 − u ) β, x = ut + (1 − u ) t ′ with β ∈ F ∗ p m , then ax = βt ′ − uβt ′ , and T r ( ax ) = tr ( βt ′ ) − utr ( βt ′ ) . Taking Gray mapyields φ ( ev ′ ( a )) = ( tr ( βt ′ ) , tr ( βt ′ )) t,t ′ . Takingcharacter sums θ ( a ) = 2 P t ∈ F ∗ pm P t ′ ∈ F ∗ pm ω tr ( βt ′ ) =2 − p m . Thus w L ( ev ′ ( a )) = 2( p − p m − − p m − ) . (d) Let a = uα + (1 − u ) β ) , x = ut + (1 − u ) t ′ . So ax = βt ′ + u ( αt − βt ′ ) , and T r ( ax ) = tr ( βt ′ ) + utr ( αt − βt ′ ) . Taking Gray map yields φ ( ev ′ ( a )) = (cid:0) − tr ( αt − βt ′ ) , tr ( αt + βt ′ ) (cid:1) t,t ′ . Taking charactersums θ ( a ) = P t ∈ F ∗ pm ω − tr ( αt ) P t ′ ∈ F ∗ pm ω tr ( βt ′ ) + P t ∈ F ∗ pm ω tr ( αt ) P t ′ ∈ F ∗ pm ω tr ( βt ′ ) = 2 . Thus w L ( ev ′ ( a )) = p − p ( N ′ − , or w L ( ev ′ ( a )) = 2( p − p m − − p m − ) . Thus we have constructed a p -ary code of length N ′ = 2( p m − , dimension m, with two weights.The weight distribution is given in Table III. Note thatthe parameters are different from those in [4], [14],[15] and [16]. Table III . weight distribution of C ′ ( m, p ) inTheorem 5.5Weight Frequency0 1 p − p m − − p m − ) p m − p m + 12( p − p m − − p m − ) 2 p m − VI. O
PTIMALITY OF THE p - ARY IMAGE
A central question of coding theory is to decidewhether the constructed codes are optimal or not. Inthis section, we will investigate the optimality of the p -ary image of the trace codes we constructed in Section OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 2017 5
V. Recall the p -ary version of the Griesmer bound. If [ N, K, d ] are the parameters of a linear p -ary code,then K − X j =0 (cid:24) dp j (cid:25) ≤ N. A. L = u Q + (1 − u ) F ∗ p m , m is odd and p ≡ Theorem 6.1
Assume m is odd and m ≥ , and p ≡ . The code φ ( C ( m, p )) is optimal. Proof
Firstly, N = p m − p m + 1 , K = 2 m, d =( p − p m − − p m − ) on account of Theorem5.4. We claim that K − P j =0 (cid:6) d +1 p j (cid:7) > N, contradictingthe Griesmer bound. The ceiling function takes threevalues depending on j . • ≤ j ≤ m − ⇒ ⌈ d +1 p j ⌉ = p m − j − p m − j − p m − j − + 2 p m − j − + 1 ; • j = m ⇒ ⌈ d +1 p j ⌉ = p m − p m − − ; • m < j ≤ m − ⇒ ⌈ d +1 p j ⌉ = p m − j − p m − j − .Thus K − P j =0 (cid:6) d +1 p j (cid:7) = p m − p m + m. Note that K − P j =0 ⌈ d +1 p j ⌉ − N = m − > . Example 6.2
Let p = 3 and m = 3 , we obtain aternary code of parameters [676 , , . The weightsof this code are 450 and 468 with frequencies 676 and52, respectively. B. L ′ = R ∗ , m is an arbitrary integer and p is oddprime Theorem 6.3
Assume m ≥ and p = 3 or m ≥ and the odd prime p ≥ . The code φ ( C ′ ( m, p )) isoptimal. Proof
It follows from Theorem 5.5 that N ′ =2( p m − p m + 1) , K = 2 m, d = 2( p − p m − − p m − ) . We claim that K − P j =0 (cid:6) d +1 p j (cid:7) > N ′ , violatingthe Griesmer bound. The ceiling function takes thefollowing values depending on the position of j .(a) when p = 3 , the ceiling function takes threevalues depending on j . • ≤ j ≤ m − ⇒ ⌈ d +1 p j ⌉ = 2 p m − j − p m − j − p m − j − + 4 p m − j − + 1 ; • j = m ⇒ ⌈ d +1 p j ⌉ = 2 p m − p m − − ; • m < j ≤ m − ⇒ ⌈ d +1 p j ⌉ = 2 p m − j − p m − j − .Thus K − P j =0 (cid:6) d +1 p j (cid:7) = 2 p m − p m + m. Note that K − P j =0 (cid:6) d +1 p j (cid:7) − N ′ = m − > . (b) when p ≥ and p is odd prime, the ceilingfunction takes three values depending on j . • ≤ j ≤ m − ⇒ ⌈ d +1 p j ⌉ = 2 p m − j − p m − j − p m − j − + 4 p m − j − + 1 ; • j = m ⇒ ⌈ d +1 p j ⌉ = 2 p m − p m − − ; • m < j ≤ m − ⇒ ⌈ d +1 p j ⌉ = 2 p m − j − p m − j − .Thus K − P j =0 (cid:6) d +1 p j (cid:7) = 2 p m − p m + m − . Notethat K − P j =0 (cid:6) d +1 p j (cid:7) − N ′ = m − > . Hence the theorem is proved.
Example 6.4
Let p = 3 and m = 3 , we obtain aternary code of parameters [1352 , , . The weightsof this code are 900 and 936 with frequencies 676 and52, respectively.VII. T
HE MINIMUM DISTANCE OF THE DUAL CODE
We compute the dual distance of φ ( C ( m, p ))(resp . φ ( C ′ ( m, p ))) . In connectionwith the discussion in [14], we mention without proofthe following lemma.
Lemma 7.1
If for all a ∈ R , we have that T r ( ax ) =0 , then x = 0 . Theorem 7.2
For all odd primes p and all m ≥ , the dual Lee distance d ′ of C ( m, p ) is . Proof
First, we check that d ′ ≥ by showing that C ( m, p ) ⊥ does not contain a word of Lee weightone. If it does, let us assume first that it has value γ = 0 at some x ∈ L. This implies that ∀ a ∈R , γT r ( ax ) = 0 or T r ( aγx ) = 0 , and by Lemma7.1 γx = 0 . Contradiction with γ = 0 . If that wordtakes the value γ (1 − u ) at some x ∈ L, then writing x = ut + (1 − u ) t ′ and a = uα + (1 − u ) β, with α, β in F p m , with t ∈ Q , t ′ ∈ F ∗ p m yields, afterreduction modulo M the equation valid ∀ α, β ∈ F p m ,γtr ( βt ′ ) = 0 and tr ( γαt ) = 0 , and we conclude, bythe nondegenerate character of tr () , that γt = γt ′ = 0 . Contradiction with x = 0 .Next, we shall show that d ′ < . If not, we can applythe sphere-packing bound to φ ( C ( m, p )) ⊥ , to obtain p m ≥ N ( p −
1) = 1 + ( p m − p m + 1)( p − , or,after expansion p m + 2 p m +1 ≥ p + p m ( p m +1 + 2) . Dropping the p in the RHS, and dividing both sidesby p m , we find that this inequality would imply p m Theorem 7.3 For all odd prime p and all m ≥ , the dual Lee distance d ′ of C ′ ( m, p ) is . Proof First, we check that d ′ ≥ . The proof of d ′ ≥ is like that in Theorem 7.2. Next, we show that d ′ < . Otherwise, we can apply the sphere-packingbound to φ ( C ′ ( m, p ) ⊥ ) , to obtain p m ≥ N ′ ( p − 1) = 1 + 2( p m − p m + 1)( p − . OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 2017 6 It is shown by a straightforward argument that N ′ ( p − > N ′ where p is an odd prime. Then wecheck p m > N ′ . We obtain the inequality ( p m − p m − < after calculation which is contradict p m ≥ . Hence p m ≤ N ′ ( p − 1) = 1 + 2( p m − p m + 1)( p − . VIII. A PPLICATIONS TO SECRET SHARINGSCHEMES To illustrate an application of our constructed codesin secret sharing, we review the fundamentals of thiscryptographic protocol. The concept of secret sharingschemes was first proposed by Blakley and Shamir in1979. We present some basic definitions concerningsecret sharing schemes and refer the interested readerto the survey [17] for details.The sets of participants which are capable of re-covering the secret S are called access sets . The setof all access sets is called the access structure of thescheme. An access set is called minimal if its memberscan recover the secret S but the members of any ofits proper subsets cannot recover S . Furthermore, if aparticipant is contained in every minimal access set inthe scheme, then it is a dictatorial participant.The support s ( x ) of a vector x in F Np is definedas the set of indices where it is nonzero. We saythat a vector x covers a vector y if s ( x ) contains s ( y ) . A minimal codeword of a linear code C isa nonzero codeword that does not cover any othernonzero codeword.Next, we recall the secret sharing scheme based onlinear codes which is constructed by Massey. Let C be an [ n, k ] linear code over the given finite field F p and G = [ g , g , · · · , g n − ] be a generator matrix of C where the column vectors are nonzero. The dealerchooses a random vector u = ( u , u , · · · , u k − ) ∈ F kp and encodes the chosen vector as c = uG =( c , c , · · · , c n − ) . Then the dealer keeps the value of u at the first coordinate S = c = ug as a secret , anddistributes the values at the remaining coordinates of c to the participants as shares . Note that S = c = ug , the set of shares { v i = c i , · · · , v i t = c i t } can recoverthe secret S if and only if the vectors g , g i , · · · , g i t are linearly independent. Hence, we can write thesecret as a linear combination S = ug = t X j =1 x j c i j . From this equation, it is clear that if we have shares c i j , ≤ j ≤ t and find x j by g = P tj =1 x j g i j ,we can recover the secret S. In fact, we can usethe participants that correspond to the nonzero coor-dinates of the minimal codewords v ∈ C ⊥ , because cv = c + c i v i + · · · + c i t v i t which implies that S = c = − ( c i v i + · · · + c i t v i t ) . In general, it is a tough task to determine theminimal codewords of a given linear code. However,there is a numerical condition, derived in [1], bearingon the weights of the code, that is easy to check. Lemma 8.1 (Ashikmin-Barg) Let w and w ∞ de-note the minimum and maximum nonzero weights, re-spectively. If w w ∞ > p − p , then every nonzero codewordof C is minimal.In the special case when all nonzero codewords areminimal, it was shown in [7] that there is the followingalternative, depending on d ′ . Lemma 8.2 ([7]) Let C be an [ n, k ] code over F p and G = [ g , g , · · · , g n − ] be a generator matrix of C .If every codeword of C is minimal vector, then thereare p k − minimal access sets and the total number ofparticipants is n − in the secret sharing scheme basedon C ⊥ . Let d ′ denote the minimal distance of C ⊥ . Wehave the following results: • If d ′ ≥ , then for any fixed ≤ t ′ ≤ min { k − , d ′ − } , every group of t ′ participants is involvedin ( p − t ′ p k − ( t ′ +1) out of p k − minimal accesssets. We call such participants dictators. • When d ′ = 2 , if g i is a multiple of g , ≤ i ≤ n − , then the participant P i must be in everyminimal access set and such a participant is calleda dictatorial participant. If g i is not a multiple of g , then the participant P i must be in ( p − p k − out of p k − minimal access sets. Theorem 8.3 Let G = [ g , g , ..., g ( p m − − ] be agenerator matrix of φ ( C ( m, p )) where m ( ≥ is oddand p ≡ , then there are p m − minimalaccess sets and the total number of participants is ( p m − − in the secret sharing scheme based on φ ( C ( m, p ) ⊥ ) . We have the following results: • if g i is a multiple of g , ≤ i ≤ ( p m − − ,then the participant P i must be in every minimalaccess set and such a participant is called adictatorial participant. • If g i is not a multiple of g , then the participant P i must be in ( p − p m − out of p m − minimalaccess sets. Proof By the preceding lemma with w = ( p − p m − − p m − ) and w ∞ = ( p − p m − − p m − ) in Table II. Rewriting the inequality of the lemma as pw > ( p − w ∞ , and dividing both sides by p m p − , weobtain p ( p m − > ( p − p m − , or p m − p − > , which is true for m ≥ . Substitute n = ( p m − into Lemma 8.2, and then the conclusion is obtained. Remark 8.4 By the similar method in the proof ofTheorem 8.3, we find that all the nonzero codewords of φ ( C ( m, p )) ( m ( ≥ is singly-even) and φ ( C ′ ( m, p )) ( m ≥ , p is odd prime) are minimal. Thus, when m ( ≥ is singly-even, the secret sharing scheme basedon φ ( C ( m, p ) ⊥ ) has a similar structure as that in thecase when m ≥ is odd and p ≡ . Let OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 2017 7 G ′ = [ g , g , ..., g p m − − ] be a generator matrix of φ ( C ′ ( m, p )) , then there are p m − minimal access setsand the total number of participants is p m − − in the secret sharing scheme based on φ ( C ′ ( m, p ) ⊥ ) . If g j is a multiple of g , ≤ j ≤ p m − − , thenthe participant P j must be in every minimal access setand such a participant is called a dictatorial participant.If g j is not a multiple of g , then the participant P j must be in ( p − p m − out of p m − minimal accesssets. IX. C ONCLUSION In the present paper, we have studied two infinitefamilies of trace codes defined over a finite ring.Because the defining sets of these codes have thestructure of abelian multiplicative groups, they inheritthe structure of abelian codes. It is an open problemto determine if they are cyclic codes or not. Moreimportantly, it is worthwhile to study other definingsets that are subgroups of the group of units of R . In particular, it would be interesting to replace ourGaussian periods Q, N by other character sums thatare amenable to exact evaluation, in the vein of thesums which appear in the study of irreducible cycliccodes [6], [13]. This would lead to other enumerativeresults of codes with a few weights.Compared to the codes we constructed by similartechniques in [14], [15] and [16], the obtained codesin this paper have different weight distributions. Theyare also different from the weight distributions in theclassical families in [4]. Hence, the p -ary linear codesconstructed from trace codes over rings in this paperare new. Biographies of all authors Minjia Shi, he received the B.S. degree in Mathemat-ics from Anqing Normal College of China in 2004; theM.S. degree in Mathematics from Hefei University ofTechnology of China in 2007, and the Ph.D degree inthe Institute of Computer Network Systems from HefeiUniversity of Technology of China in 2010. He hasbeen teaching at the School of Mathematical Sciencesof Anhui University since 2007. Since April 2012, hehas been the Associate Professor with the School ofMathematical Sciences, Anhui University of China.His research interests include algebraic coding the-ory and cryptography. He is the author of more than 60journal articles and of one book. He has been the AreaEditor of Journal of Algebra Combinatorics DiscreteStructure and Application since 2014.He has held visiting positions in School of Physical& Mathematical Sciences, Nanyang TechnologicalUniversity, Singapore, from August 2012 to August2013, Telecom ParisTech, Paris, from July 2016to August 2016, Chern Institute of Mathematics inNankai University since 2013. Yue Guan, she has been an undergraduate studentat the School of Mathematical Sciences of AnhuiUniversity since 2013. She won Honorable Mentionof Mathematical Contest in Modeling Certificate ofAchievement in 2016 and gained scholarships severaltimes during school years. She will become a graduatestudent at School of Mathematical Sciences of AnhuiUniversity in 2017. Her research direction includescoding theory and cryptography.Patrick Sol´e received the Ing´enieur and Docteur-Ing´enieur degrees both from Ecole NationaleSup´erieure des T´el´ecommunications, Paris, France,in 1984 and 1987, respectively, and the habilitationdiriger des recherches from Universit´e de Nice-SophiaAntipolis, Sophia Antipolis, France, in 1993.He has held visiting positions in Syracuse Univer-sity, Syracuse, NY, from 1987 to 1989, MacquarieUniversity, Sydney, Australia, from 1994 to 1996, andLille University, Lille, France, from 1999 to 2000.Since 1989, he has been a permanent member of theCNRS and became Directeur de Recherche in 1996.He is currently member of the CNRS lab LAGA, fromUniversity of Paris 13.His research interests include coding theory (codesover rings, quasi-cyclic codes), interconnection net-works (graph spectra, expanders), vector quantization(lattices), and cryptography (boolean functions, pseudorandom sequences).He is the author of more than 150 journal articlesand of three books.He was the associate editor of the IEEE InformationTheory Transactions from 1996 till 1999. 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