New completely regular q-ary codes based on Kronecker products

1 Ne w completely re gular q -ary codes based on Kronecke r products J. Rif ` a, V .A. Zino viev Abstract For any integer ρ ≥ 1 and for any prim e power q , th e explicit constru ction of a infinite family of completely r egular (an d co mpletely tran siti ve) q - ary codes with d = 3 and with covering radius ρ is giv en. T he intersection array is also computed. Under the same conditions, the explicit con struction of an infinite family of q -a ry uniform ly packed codes (in the wid e sense) with covering ra dius ρ , which are not completely regular, is also giv en. In both constructions the Kronecker produc t is the ba sic tool that has been used. Index T erms Completely regular cod es, completely transitive cod es, covering radius, Kron ecker p roduct, inter- section numbe rs, u niform ly pac ked co des. I . I N T RO D U C T I O N Let F q be a finite field of t he order q . Let wt ( v ) denote the Hammi ng weight of a vector v ∈ F n q and let d ( v , u ) = wt ( v − u ) denote the Hammin g distance between two vectors v , u ∈ F n q . W e say that two vectors v and u are neighbo rs if d ( v , u ) = 1 . A q -ary linear [ n, k, d ] q -code C is a This work has been partially supported by the Spanish MEC and the European FEDER Grants MTM200 6-03250 and TSI2006- 14005-C02 -01 and also by the Russian fund of fundamental researches (the number of project 06 - 01 - 00226). Part of the material in Section III was presented at the 2nd International Castle Meeting on Coding Theory and Applications (2ICMCT A), Medina del Campo, S pain, September 2008. Dept. of Information and Communications Engineering, Uni ve rsitat Aut ` onoma de Barcelona, 0819 3-Bellaterra, Spain Institute for Problems of Information T ransmission of the Russian Academy of Sciences, Bol’ shoi Karetn yi per . 19, GSP-4, Mosco w , 101447, Russia 2 k -dimens ional sub space of F n q , where n is the length , N = q k is the car dinality of C and d is the minimum dista nce , d = min { d ( v , u ) : v , u ∈ C , v 6 = u } . The error correcting capabil ity of a code C with m inimum distance d i s given by e = ⌊ ( d − 1) / 2 ⌋ . Giv en any vector v ∈ F n q , its dist ance t o the code C is d ( v , C ) = min x ∈ C { d ( v , x ) } and the covering radius of the code C is ρ = max v ∈ F n q { d ( v , C ) } . Let D = C + x be a coset of C , where + means the comp onent-wise addit ion in F q . The weight wt ( D ) of D is t he m inimum weig ht of the codewords of D . For an arbitrary coset D of C of weight s = wt ( D ) denote by µ ( D ) = ( µ 0 ( D ) , µ 1 ( D ) , ..., µ n ( D )) its weight distribution, where µ j ( D ) , j = 0 , . . . , n denotes the num ber of words of D of weigh t j . N otice that µ j ( D ) = 0 for all j < s . Definition 1: A q -ary linear code C with covering radius ρ is called completely r e gular if the weight dis tribution of any coset D of C of weigh t i , i = 0 , 1 , ..., ρ is uniquely defined by the minimum weight of D , i.e. by the num ber i = wt ( D ) . Definition 2: Let C b e a q -ary code o f l ength n and let ρ be its covering radius. W e say that C is unifo rmly packed in the wide sense, i.e. in t he s ense o f [1], if there exist rational numbers α 0 , . . . , α ρ such that for any v ∈ F n q ρ X k =0 α k f k ( v ) = 1 , (1) where f k ( v ) i s the num ber of codew ords at distance k from v . The case ρ = e + 1 correspond s to uniformly packe d codes , suggested in [10], and th e case ρ = e + 1 and α ρ − 1 = α ρ corresponds to uniforml y pack ed codes in the narr ow sense or sometimes called str o ngly u niformly packe d codes , sug gested in [18]; s ee more special cases of such codes in [7], [9], [10], [18]. It is well known (see, for example, [5]) that any completel y regular code is un iformly packed in the wide sense. In turn, uniform ly packed codes with ρ = e + 1 are completely regular [10], [18], includin g some extended such codes [1], [2], [18]. But til l now , the only known examples of u niformly packed codes, whi ch are not completely regular , were the known binary (primit iv e in narrow sense) BCH codes o f leng th n = 2 m − 1 ( m odd) with 3 minimum dist ance d = 7 [6] and the Z 4 -linear Goeth als-like codes of length n = 2 m − 1 ( m e ven) wit h min imum di stance d = 7 [12] (includi ng extended codes for both families of codes). In bot h cases the codes have covering radius ρ = e + 2 = 5 , and ρ = e + 3 = 6 for extended codes. It has been conjectured for a lo ng ti me that i f C is a completely regular code and | C | > 2 , then e ≤ 3 . For th e special case of li near comp letely transitive codes [16], the analogous con jecture was solved in [3] and [4] proving that for e ≥ 4 such n ontrivial codes do not exist. Hence, the existing completely regular codes and com pletely transi tiv e codes have an smal l error correcting capability . In respect of the covering radius, Sol ´ e in [16] uses the direct sum of ℓ copies of fixed perfect binary 1 -code of length n to cons truct infinite families of binary completely regular codes of leng th n · ℓ with covering radius ρ = ℓ . Thus, using [16], the covering radius of the resulting code is growing to infinity with th e length of the code. One of the main purpose in t he current paper is t o describe a method of constructing linear completely re gular and completely transitive codes with arbitrary covering radius, which is constant wh en t he l ength of the resulting code is growing to infinit y . M ore exactly , for any prime power q and for any natu ral nu mber ℓ we giv e, in T heorem 1, an explicit construction of an infinite family of linear q -ary completely regular and completely transitive codes with lengt hs n = ( q m − 1)( q ℓ − 1) / ( q − 1) 2 and with fixed covering radius ρ = ℓ , wh ere m ≥ ℓ is any integer (a previous approach in t his direction can be found in [15]). The intersection array for t hese completely regular codes is computed i n Theorem 2. Under the same conditions (i.e. for any prime power q and for any natural number ℓ ) we give the explicit construction of an infinite family of q -ary l inear uniformly p acked codes (in the wide sense) with lengt hs n = ( ℓ + 1) ( q m − 1) / ( q − 1) and wit h covering radius ρ = ℓ , where m, ℓ ≥ 2 are any integers. All t hese codes (wi th the exception q = ℓ = 2 ) are not completely regular . I I . P R E L I M I N A RY R E S U L T S For a giv en q -ary code C with covering radius ρ = ρ ( C ) define C ( i ) = { x ∈ F n q : d ( x , C ) = i } , i = 0 , 1 , . . . , ρ. W e also use the following alternativ e stand ard definition of complet ely regularity [14]. 4 Definition 3: A code C is completely regular , if for all l ≥ 0 e very vector x ∈ C ( l ) has the same number c l of neighb ors in C ( l − 1) and the same number b l of neighb ors in C ( l + 1) . Also, define a l = ( q − 1) · n − b l − c l and note that c 0 = b ρ = 0 . Refer to ( b 0 , . . . , b ρ − 1 ; c 1 , . . . , c ρ ) as the intersection array o f C . For a q -ary [ n, k , d ] q -code C with weight dis tribution µ ( C ) = ( µ 0 , . . . , µ n ) define the o uter distance s = s ( C ) as the number o f non zero coordinates µ ⊥ i , i = 1 , . . . , n of the vector ( µ ⊥ 0 , . . . , µ ⊥ n ) ob tained by the MacW illiams transform of µ ( C ) [8]. Hence, since C i s a linear code, s ( C ) i s the num ber of differe nt non zero weights of codewords in th e dual code C ⊥ . Lemma 1 ([8]): For any code C with cov ering radius ρ ( C ) and wit h outer distance s ( C ) we hav e ρ ( C ) ≤ s ( C ) . Lemma 2: Let C be a code with mi nimum distance d = 2 e + 1 , covering radius ρ and ou ter distance s . Then: 1) Code C i s uniformly packed in the wide sense if and only if ρ = s ([2]). 2) If C i s completely regular then it is un iformly packed in the wi de sense ([5]). 3) If C is uniformly packed in the wi de sense and ρ = e + 1 , then it is completely regular ([18], [10]). Let C be a linear code of length n over F q , a finite fie ld of size a prime powe r q . Following [13], if q = 2 , the autom orphism group Aut ( C ) of C is a subgroup o f the sym metric group S n consisting of all n ! permutations of the n coordinate posi tions which send C into itself. Let M be a monomial matrix, i.e. a matrix with exactly o ne no nzero entry in each row and column. If q i s prime, then Aut ( C ) consist s of all n × n m onomial matrices M over F q such that c M ∈ C for all c ∈ C . If q is a power of a prim e n umber , then Au t ( C ) also contains all the field automorphi sms o f F q which preserve C . The grou p Aut ( C ) i nduces an acti on on the set of cos ets of C in the following way: for all φ ∈ Aut ( C ) and for every vector v ∈ F n q we hav e φ ( v + C ) = φ ( v ) + C . In [16] it was introduced the concept of completely transitive bi nary li near code and it can be generalized to the fol lowing definition, which also corresponds to the definitio n of coset- completely transitive code in [11]. Definition 4: Let C be a linear code over F q with covering radius ρ . Then C i s completely transitive if Aut ( C ) has ρ + 1 orbi ts when acts on the cosets of C . 5 Since two cosets in th e same orbit should have the same wei ght distribution, it is clear that any com pletely transitive code is completely regular . I I I . K RO N E C K E R P RO D U C T C O N ST RU C T I O N In this sectio n we describe a n e w constructi on which provides for any natural number ρ and for any prim e power q an infinite family of q -ary l inear com pletely regular codes with covering radius ρ . Definition 5: For two m atrices A = [ a r,s ] and B = [ b i,j ] over F q define a ne w matrix H which is the Kronecker product H = A ⊗ B , where H is obtain ed by changing any element a r,s in A by the matrix a r,s B . Consider the matrix H = A ⊗ B and let C , C A and C B be the codes over F q which have , respectiv ely , H , A and B as a parity check matrices. Assum e t hat A and B hav e size m a × n a and m b × n b , respectively . For r ∈ { 1 , · · · , m a } and s ∈ { 1 , · · · , m b } the rows in H l ook as ( a r, 1 b s, 1 , · · · , a r, 1 b s,n b , a r, 2 b s, 1 , · · · , a r, 2 b s,n b , · · · , a r,n a b s, 1 , · · · , a r,n a b s,n b ) . Arrange these rows taki ng blocks of n b coordinates as colum ns such that the vectors c in code C are presented as matrices of size n b × n a : c =        c 1 , 1 . . . c 1 ,n a c 2 , 1 . . . c 2 ,n a . . . . . . . . . c n b , 1 . . . c n b ,n a        =        c 1 c 2 . . . c n b        , (2) where c i,j = a r,j b s,i and c r denotes the r -th row vector of this m atrix. W e will call matrix representation th e above way to present th e vectors c ∈ C . Let us go to a further v iew on the code words of C , t he code ov er F q which has H = A ⊗ B as a parity check matrix. Consider vector c ∈ C and use t he representation in (2), hence c = ( c 1 , c 2 , · · · , c n b ) t , where ( · ) t means the transpose vector . No w compute th e syndrome vector which leads us to a ( m b × m a ) matrix that we will equal to zero. W e have B  A c t 1 , A c t 2 , . . . , A c t n b  t = 0 and so, B  A c t  t = B · c · A t = 0 . (3) W ith t his last property it is easy to note that any ( n b × n a ) matrix with codew ords of C A as rows belong to the code C and also any ( n b × n a ) matrix with codewords of C B as columns b elongs 6 to the code C . V ice versa, all the codew ords in C can alw ays be seen as linear com binations of matrices of both ty pes above. Moreover , it is straightforward to state the following well kn own fact. Lemma 3: Codes d efined by the parity check matrices A ⊗ B and B ⊗ A are permutation equiv alent . From no w on, we assume that matrix A (respecti vely , B ) is a parity check matrix of a Hamming code with parameters [ n a , k a , 3] q (respectiv ely , [ n b , k b , 3] q ), where n a = ( q m a − 1) / ( q − 1) ≥ 3 (respectiv ely , n b = ( q m b − 1) / ( q − 1) ≥ 3 ) and k a = n a − m a (respectiv ely , k b = n b − m b ). Denote by H m the parity check matri x of a perfect Hamming [ n, k , 3 ] q -code C over F q , where n = ( q m − 1) / ( q − 1) . Let ξ 0 = 0 , ξ 1 = 1 , . . . , ξ q − 1 denote the elements of F q . Then t he m atrix H m can be expressed, up to equivalence , through the matrix H m − 1 as follows [17]: H m =   0 · · · 0 1 · · · 1 · · · ξ q − 1 · · · ξ q − 1 1 H m − 1 H m − 1 · · · H m − 1 0   , where 0 is the zero column and where H 1 = [1] . Note that, under such construction, the following lemmas are straight forward (see, for example, [17]). Lemma 4: Matrix H m contains as columns , amo ng other , all the m possibl e binary vectors of length m and o f weight 1 . Lemma 5: For i = 1 , . . . , m , let r i denote the i -th row of H m . Let g = P m i =1 ξ i r i , with ξ i ∈ F q , be any linear comb ination of the rows o f H m . If wt ( g ) 6 = 0 , t hen wt ( g ) = q m − 1 . Throughout this work we wil l consid er the colum ns in A and B ordered in such a way that the one-weighted vectors will be placed i n the first m a (respectiv ely , m b ) positi ons. Any codeword c ∈ C , which has nonzero elements o nly in o ne row (or only in one col- umn) w ill be called a line . Since A and B are parity check matrices of Hamm ing codes (i.e. they have m inimum distances 3 ), there are li nes of weight 3 . For example, a row line L r = ( α 1 , α 2 , α 3 ) ( s 1 ,s 2 ,s 3 ) (respectiv ely , a colum n line L s = ( α 1 , α 2 , α 3 ) ( r 1 ,r 2 ,r 3 ) ) means th at the code word c of weight 3 , whose nonzero r th ro w (respectively , nonzero s t h colum n) has nonzero elements α 1 , α 2 , α 3 in col umns s 1 th, s 2 th, s 3 th (respectiv ely , in rows r 1 th, r 2 th, r 3 th). Recall that t his means the foll owing equality for the correspondi ng colu mns a s 1 , a s 2 , and a s 3 of matrix 7 A (respectiv ely , for the columns b r 1 , b r 2 , and b r 3 of matrix B ): 3 X i =1 µ i a s i = 0 ( respectively , 3 X j =1 λ j b r j = 0 ) . (4) Define th e set of row indices as R = { 1 , . . . , n b } (respectively , of column i ndices as S = { 1 , . . . , n a } ) and assume that the first m b indices (respectiv ely , th e first m a ) corresponds to the column vectors in A (respectiv ely , i n B ) of weight one. By definition of perfect codes, for a fixed row index r ∈ R (respectively , column index s ∈ S ), for any two no nzero elements α 1 , α 2 ∈ F q and for any two differe nt s 1 , s 2 ∈ S (respectively , r 1 , r 2 ∈ R ) there is a u nique row line L r = ( α 1 , α 2 , α 3 ) ( s 1 ,s 2 ,s 3 ) (respectiv ely , column li ne L s = ( α 1 , α 2 , α 3 ) ( r 1 ,r 2 ,r 3 ) ) for some nonzero element α 3 ∈ F q and for some s 3 ∈ S (respectiv ely , r 3 ∈ R ). It i s well known that th e li near span of the vectors of weight three in a Hamming code gives all the code. Hence, th e linear sp an of the row lines of weight three and the column lines of weight three gives all the cod e words of C . Giv en a vector v ∈ F n b · n a q let v = [ v ij ] be i ts matrix representatio n. W e will call m ain submatrix the ( m b × m a ) matrix contain ing the first m b rows and m a columns of t he matrix representation. It is easy to see t hat, after simpli fying (i.e. passing lin es t hrough t he p oints p laced out of the m ain submatrix), we can obtain a ne w vector v ′ in the same coset v + C such that its matrix representation has zero elements everywhere except into the main su bmatrix M v . Lemma 6: Let v ∈ F n b · n a q be a vector and l et M v be its main submatrix representation. Then: 1) V ector v is in C if and on ly if M v = 0 . 2) For each v the main submatri x representation M v is unique. Pr oof : First o f all, take a nonzero ( m b × m a ) matrix M . Each column (respectively , ro w) is not a line, indeed, we would have a line L r = ( α 1 , . . . , α m a ) s 1 ,...,s m a in volving only i ndependent vectors o f weigh t o ne, which is impos sible. Hence, t he conclu sion is that it is i mpossibl e that such a n onzero main submatrix M is a code word. V ice versa, given a vector v ∈ C and doing the simplification operations described above we will obtain a zero main submatri x representation. The second p oint is a coroll ary of the first one. Giv en a vector v ∈ F n b · n a q let v = [ v ij ] be its matrix representati on. Compute the syndrome S v like i n (3) which is a ( m b × m a ) matrix. Note that adding ( n a − m a ) zero columns and 8 ( n b − m b ) zero ro ws to t his syndrom e matrix we obtain the above main sub matrix representation M v for v . Hence, in ot her words: Lemma 7: Given a vector v ∈ F n b · n a q let v = [ v ij ] be it s matrix representation. Then: ( A ⊗ B )( v ) = S v = B [ v ij ] A t = B M v A t Consider m b ≥ m a (in t he contrary case we wil l do t he same b ut reverting the role of matrices A and B ). T ake a vector e ∈ F n b · n a q such that all the elements in th e matrix representation are zeroes, except o ne. So, there are two specific v alues 1 ≤ λ ≤ n b , 1 ≤ µ ≤ n a such that e = [ e ij ] ; e λµ = e and e ij = 0 for all i 6 = λ and j 6 = µ . Using (4), we can pass a colum n line across the point ( λ, µ ) o btaining one or m ore alig ned points in the first m b rows. Again, passing row li nes across these last points we obt ain the main submatrix representation which i s as follows: M µ ⊗ λ = e ·        µ 1 λ 1 µ 2 λ 1 · · · µ m a λ 1 µ 1 λ 2 µ 2 λ 2 · · · µ m a λ 2 . . . . . . . . . . . . µ 1 λ m b µ 2 λ m b · · · µ m a λ m b        , (5) where a µ = P m a i =1 µ i a s i ; b λ = P m b i =1 λ i b r i and a s i , b r i are t he on e weigh ted vectors of length m a and m b , respectively . Remark 1: Note that t he first nonzero indexes in { µ 1 , µ 2 , · · · , µ m a } and { λ 1 , λ 2 , · · · , λ m b } are µ f µ = 1 and λ f λ = 1 , respective ly . It is i mportant to poin t out that give n a ( m b × m a ) matrix M th e r ank ( M µ ⊗ λ + M ) di ff ers from r ank ( M ) in one uni t, at the mos t. Pr opo sition 1: L et v ∈ F n b · n a q be a vector and M v be its main subm atrix representation. Th en the distance of v to cod e C is d ( v , C ) = r an k ( M v ) . Pr oof : Let r ank ( M v ) = s . Doing simp lifications passing lines across the rows of M v we will obtain a representation vector wi th n onzero elements in, at maximum , s column s. Ag ain passing lines across these col umns we ob tain a representati on matrix for the giv en vector v wi th not more that s nonzero coordinates. Hence, d ( v , C ) ≤ s . 9 Now , we are going to p rove that s ≤ d ( v , C ) . Consider the vector c ∈ C with the s ame coordinates as v and, moreover the ne w d ( v , C ) coordin ates that we need to add to v to obtain that vector c in C . For each one of the coordinates v ij in which v and c differ we do the sam e consideration as in (5) and so, we see that t he rank of the main submatrix representation of v + v ij diffe rs from the previous in one unit, at the most. T hat is, after adding all t he necessary coordin ates to v to obtain c , t he rank of the main submatrix representati on varied in, at the most, d ( v , C ) units obtaining the final value of zero. Hence, t he i nitial rank s must be necessarily less or equal to d ( v , C ) . The fol lowing theorem shows that the code cons tructed by the Kronecker product is a com- pletely transitive code and, therefore, is a comp letely regular code. Theor em 1: Let C be t he code ove r F q which has H = A ⊗ B as a parity check matrix, where A and B are parity check m atrices of Hamming codes [ n a , k a , 3] q and [ n b , k b , 3] q , respective ly , where n a = ( q m a − 1) / ( q − 1) ≥ 3 ; n b = ( q m b − 1) / ( q − 1) ≥ 3 ; k a = n a − m a and k b = n b − m b . Then: 1) Code C h as length n = n a · n b , dimensi on k = n − m a · m b and mini mum distance d = 3 . 2) The covering radius of C is ρ = min { m a , m b } . 3) Code C i s completely transit iv e and, th erefore, a comp letely regular code. Pr oof : It is straightforward to check that the code C has length n = n a · n b , dim ension k = n − m a · m b and mini mum distance d = 3 . In respect of the covering radius, take a vector v ∈ F n b · n a q and use Proposition 1. Matrix M v is a ( m b × m a ) matrix, so this rank is an integer value from 0 to min ( m a , m b ) . T o prove that C is a completely transitive code it is enough to show th at starting from two vectors x , y ∈ C ( ℓ ) , there exists a monomi al matrix φ ∈ Aut ( C ) such th at φ ( x ) ∈ y + C or , in other words, ( A ⊗ B )( φ ( x )) = ( A ⊗ B )( y ) . First of all, l et φ 1 be any monomi al ( n a × n a ) matrix and φ 2 be any monomi al ( n b × n b ) matrix. It is clear that ( Aφ 1 ) ⊗ ( B φ 2 ) = ( A ⊗ B )( φ 1 ⊗ φ 2 ) and φ 1 ⊗ φ 2 is a monom ial ( n a n b × n a n b ) mat rix. Moreover , we hav e that Aut ( A ⊗ B ) = Aut  ( A ⊗ B ) ⊥  = Au t ( C ) and so I d ⊗ φ ∈ Aut ( C ) . Hence, if φ is an autom orphism in Aut ( B ) 10 then I d ⊗ φ ∈ Au t ( C ) . The two given vectors x , y belong to C ( ℓ ) and so, r ank ( S x ) = rank ( S y ) , w here S x and S y are the syndrome of x and y , respectively . T o prove that C is a completely transitive code we will show that there exists an automorphism φ ∈ Aut ( B ) such that ( A ⊗ B )( y ) = ( A ⊗ B φ )( x ) = ( A ⊗ B )( φ ( x )) . Assume m b ≥ m a (otherwise, we wil l do t he same construction rev erting A and B ). It is straightforward to find an in vertible ( m b × m b ) matrix K over F q such that S t x K = S t y . Since B is the parity check m atrix of a Hamming code, the matrix K t B is again a parity check matrix for a Hamm ing code and K t B = B φ for s ome monom ial matrix φ . M oreover , if G B is the corresponding generator matrix for t his Hamm ing code, i.e. B G t B = 0 , then ( B φ ) G t B = ( K t B ) G t B = 0 and so φ ∈ Aut ( B ) . Finally , ( A ⊗ B )( y ) = S y = K t S x = K t ( B x A t ) = B φ x A t = ( A ⊗ B φ )( x ) = ( A ⊗ B )( φ ( x )) . The following goal is to comput e the intersection array for this completel y regular code C . Theor em 2: Let C A and C B be two Hamming codes of parameters [ n a , k a , 3] q and [ n b , k b , 3] q , respectiv ely , where n a = ( q m a − 1) / ( q − 1) ≥ 3 ; n b = ( q m b − 1) / ( q − 1 ) ≥ 3 with dimensi on k a = n a − m a and k b = n b − m b , respectiv ely . Let A (respecti vely , B ) be a p arity check matrix for th e code C A (respectiv ely , C B ). Then the mat rix H = A ⊗ B , the Kronecker product of A and B , is a parity check matrix o f a q -ary completely regular [ n, k , d ] q -code C with covering radius ρ , where n = n a · n b , k = n − m a · m b , d = 3 , ρ = min { m a , m b } , (6) and with intersection num bers for ℓ = 0 , 1 , . . . , ρ : b ℓ = ( q − 1)  n a − q ℓ − 1 q − 1   n b − q ℓ − 1 q − 1  , c ℓ = q ℓ − 1 q − 1 q ℓ − 1 , a ℓ = ( q − 1) · n a · n b − c ℓ − b ℓ 11 Pr oof : L et x ∈ C ( ℓ ) and y = x + e , w here e is a ( n b × n a ) matrix whi ch has one nonzero position, say e λ,µ , where λ ∈ R , µ ∈ S and e λ,µ = e ∈ F ∗ q . As we said before, after doing simplifications we can al ways think that x = [ x i,j ] has ℓ non zero positions at t he main diagonal of value 1 and the correspondi ng ℓ column vectors b i (respectiv ely a j ) are linear i ndependent. Let R 1 be t he set of t hese column vectors b i (respectiv ely , let S 1 be the set of t hese column vectors a j ). The case ℓ = 0 follows immediately (any lo cation of e λ,µ contributes clearly only to the number b 0 ): a 0 = 0 , b 0 = ( q − 1) · n a · n b . Now consider the general case: 1 ≤ ℓ ≤ ρ . First of all, assu me t hat e = [ e λ,µ ] and that the vector b λ is linearly independent from the vectors i n R 1 (respectiv ely , a µ is l inearly in dependent from t he vectors in S 1 ). The only contribution i s b ℓ and so it is easy to find that: b ℓ = ( q − 1)  n a − q ℓ − 1 q − 1   n b − q ℓ − 1 q − 1  . Now , we are g oing to t he case wh ere a µ linearly depends from the set S 1 and b λ from R 1 . From (5) and Proposi tion 1 we can assume th at a µ is lin early dependent from the vectors in S 1 and also b λ from t he vectors in R 1 . So, a µ = P m a i =1 µ i a µ i ; b λ = P m b j =1 λ j b λ j and a µ i , b λ j are the one weighted vectors of lengt h m a and m b , respectively . W e want to count in how many ways the fol lowing matrix has rank ℓ − 1 : I d ℓ + M µ ⊗ λ =                 1 + µ 1 λ 1 e µ 2 λ 1 e · · · µ ℓ λ 1 e · · · µ m a λ 1 e µ 1 λ 2 e 1 + µ 2 λ 2 e · · · µ ℓ λ 2 e · · · µ m a λ 2 e . . . . . . . . . . . . . . . . . . µ 1 λ ℓ e µ 2 λ ℓ e · · · 1 + µ ℓ λ ℓ e · · · µ m a λ ℓ e µ 1 λ ℓ +1 e µ 2 λ ℓ +1 e · · · µ ℓ λ ℓ +1 e · · · µ m a λ ℓ +1 e . . . . . . . . . . . . . . . . . . µ 1 λ m b e µ 2 λ m b e · · · µ ℓ λ m b e · · · µ m a λ m b e                 (7) If λ f λ = µ f µ = 1 (see Remark 1) are such that f λ > ℓ o r f µ > ℓ then the rank of the above matrix I d ℓ + M µ ⊗ λ would be greater than ℓ − 1 . Hence, we can transform the above m atrix in 12 the following one, which has th e same rank:                 1 + P ℓ i =1 µ i λ i e µ 2 λ 1 e · · · µ ℓ λ 1 e · · · µ m a λ 1 e 0 1 · · · 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 0 · · · 1 · · · 0 − λ ℓ +1 e/λ 1 0 · · · 0 · · · 0 . . . . . . . . . . . . . . . . . . − λ m b /λ 1 e 0 · · · 0 · · · 0                 (8) It is easy t o see th at the rank of the above matri x is ℓ − 1 plus t he rank of P , where P =        1 + P ℓ i =1 µ i λ i e µ ℓ +1 λ 1 e · · · µ m a λ 1 e − λ ℓ +1 e/λ 1 0 · · · 0 . . . . . . . . . . . . − λ m b e/λ 1 0 · · · 0        (9) W e are interested in to count in how many ways the rank of P is zero. It happens when all the µ i = 0 for ℓ + 1 ≤ i ≤ m a ; λ i = 0 for ℓ + 1 ≤ i ≤ m b and 1 + P ℓ i =1 µ i λ i e = 0 . Now , n ote that when we fix a specific values for λ 1 , λ 2 , · · · , λ ℓ , wi th λ f λ = 1 we want to count how many solutions ( µ 1 , µ 2 , · · · , µ m a ) has the equation 1 + P ℓ i =1 µ i λ i e = 0 with the restrictions λ f λ = µ f µ = 1 (see Remark 1). W e know that e is any value in F ∗ q and, on th e other side, µ f µ = 1 . Hence, each s olution ( µ 1 , µ 2 , · · · , µ m a ) to the equation 1 + P ℓ i =1 µ i λ i e = 0 with t he q uoted restrictions could be transformed in to s olution of 1 + P ℓ i =1 µ i λ i = 0 without any restriction for µ f µ . Finally , given a specific values for λ 1 , λ 2 , · · · , λ ℓ , wit h λ f λ = 1 , our prob lem consist o f counting how many sol utions ( µ 1 , µ 2 , · · · , µ m a ) the equation 1 + P ℓ i =1 µ i λ i e = 0 has, without any rest riction for µ f µ . It i s easy to see that this value is q ℓ − 1 . Doing the above account for all the q ℓ − 1 q − 1 possibili ties when you cho ose the specific values for λ 1 , λ 2 , · · · , λ ℓ we reach the st atement. I V . K RO N E CK E R P RO D U C T C O N S T RU C T I O N O F U N I F O R M L Y PAC K E D C O D E S The following theorem describes t he explicit const ruction o f infinite family of q -ar y linear uniformly packed cod es (in the wide sense) with fixed covering radius ρ , where q is any prime 13 power , and where ρ ≥ 2 is an arbitrary n atural n umber . The i nteresting fact h ere is that t hese codes are not comp letely regular . Recall that a trivial q -ary repetition [ n, 1 , n ] q -code is a perfect code, if and only if q = 2 and n is odd. Theor em 3: Let C A and C B be two linear codes: the repetition [ n a , 1 , n a ] q -code C A of length n a ≥ 3 and the q -ary perfect Hamming [ n b , k b , 3] q -code C B of length n b = ( q m − 1) / ( q − 1) ≥ q + 1 , where n a ≤ n b . Let A (respectiv ely , B ) be a parity check matrix of code C A (respectiv ely , C B ). Th en t he m atrix H = A ⊗ B , the Kronecker product of A and B , is a parity check matrix of a q -ary uni formly packed (in th e wid e sense) [ n, k , d ] q -code C with cove ring radius ρ , where n = n a · n b , k = n − m · ( n a − 1) , d = 3 , ρ = n a − 1 . (10) Furthermore, code C is not completely regular with an exception for th e case q = 2 and n b = 3 . Pr oof : Let a i (respectiv ely , b j ) denotes the i -th column of A (respectiv ely , t he j -th column of B ). Remark that A is a ( n a − 1 × n a ) -matrix: A =           1 0 0 · · · 0 − 1 0 1 0 · · · 0 − 1 0 0 1 · · · 0 − 1 · · · · · · · · · 0 0 0 · · · 1 − 1           . Hence, the matrix H = A ⊗ B has a very simple structure: H =           B 0 0 · · · 0 − B 0 B 0 · · · 0 − B 0 0 B · · · 0 − B · · · · · · · · · 0 0 0 · · · B − B           , where 0 denot es the zero matrix of size m × n b . Any q -ary vector x of l ength n − k = m · ( n a − 1) can be presented as follows: x = ( x 1 | . . . | x n a − 1 ) w here x i is a q -ar y vector of length m for any i = 1 , . . . , n a − 1 . M atrix B contains as columns, up to m ultiplicative scalar , any vector over F q of length n b − k b = m . 14 Hence for any x i , i = 1 , . . . , n a − 1 there is a colum n b j i of B s uch that x t i = ξ i b j i for some ξ i ∈ F ∗ q . Since a i ⊗ b j is a col umn of H , and since x t can be written as x t = P n a − 1 i =1 ξ i b j i , we deduce that ρ ≤ n a − 1 . T o s ee that ρ ≥ n a − 1 it is enou gh to choose as a vector x a vector with all nonzero mutually d iffe rent compon ent vectors x i , i = 1 , . . . , n a − 1 . Such a choice is possible, since q m − 1 ≥ n b ≥ n a . W e concl ude that ρ = n a − 1 . Now we turn to th e o uter di stance s = s ( C ) of C (i.e. the number of different nonzero weights of code words in C ⊥ ). Matrix B is the parity check matrix of a Hamm ing code so, after Lemma 5, we conclud e that all the nonzero linear com binations of th e ro ws in A h a ve the same weight q m − 1 . Now consider any lin ear combination over F q of ro ws of H . It is easy to see, by the shape of H that the n umber of d iffe rent nonzero weights go from 2 · q m − 1 until n a · q m − 1 so, the numb er of differe nt nonzero values for the weigh t of the codewords i n t he code C ⊥ generated by t he matrix H is equal to n a − 1 . Hence, the ou ter distance s ( C ) of C i s equal to n a − 1 and so, ρ ( C ) = s ( C ) . Now , usin g Lemma 2, w e conclude that the code C is uniformly packed i n the wide sense, i.e. in the sense of [1]. T o finish the p roof we hav e o nly to show that C is not completel y regular , w ith only one exception: when A is the trivial binary repetition [3 , 1 , 3] 2 -code which, at th e same ti me, is the trivial Hamming code of length 3 . But this l ast case (i.e. t he case q = 2 and n b = 3 ) i s includ ed in Theorem 2. Hence we ha ve only to show that in all ot her cases the code C is not com pletely regular . Consider the next possible bi nary repetit ion code. When n a = 4 and ρ = 3 we have the repetition [4 , 1 , 4] 2 -code C A . Choose as the code C B the bin ary Hamming [7 , 4 , 3] 2 -code. W e claim that t he resulti ng [28 , 1 9 , 3] 2 -code C (after applyi ng Theorem 3 ) is not a comp letely regular code. Let H = A ⊗ B , i.e. H lo oks as H =      B 0 0 B 0 B 0 B 0 0 B B      , 15 where B is the following matrix: B =      1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1      . Consider two differe nt vectors x 1 and x 2 of weight 2 , whi ch belong to C (2) . Let x 1 = (1000000 | 1000000 | 000000 0 | 0000000) and x 2 = (10000 00 | 0100 000 | 000 0000 | 00000 00) . It is easy to see that both vectors x 1 and x 2 are from C (2) and we obtain i mmediately the intersection numbers c 2 = 4 for x 1 and c 2 = 2 for x 2 . Th us, code C is not com pletely regular . Clearly the same contra-example works for q = 2 and for larger values n b ≥ 5 . For the cases q ≥ 3 these above contra-examples should be sl ightly modi fied. F or the s mallest case q = 3 and n b = 3 choos e as t he code C B the Hamming [4 , 2 , 3] 3 -code with parity check matrix B and let A be a parity check matrix of the repetition [3 , 1 , 3] 3 -code C A , where F 3 = { 0 , 1 , 2 } , B =   1 1 1 0 0 1 2 1   and A =   1 0 2 0 1 2   T ake the following vectors x 1 and x 2 from C (2) : x 1 = (1000 | 2000 | 0 000) and x 2 = (1000 | 010 0 | 0000) . W e obtain the in tersection num bers c 2 = 4 for x 1 and c 2 = 2 for x 2 . Hence, the resulti ng [12 , 8 , 3] 3 -code C is not completely regular . The same contra-example works for the rest of cases q ≥ 3 and n b ≥ 3 . 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