Homogeneous Weights of Matrix Product Codes over Finite Principal Ideal Rings
aa r X i v : . [ c s . I T ] A p r Homogeneous Weights of Matrix Product Codesover Finite Principal Ideal Rings
Yun Fan , San Ling , Hongwei Liu School of Mathematics and Statistics, Central China NormalUniversity, Wuhan 430079, China Division of Mathematical Sciences, School of Physical &Mathematical Sciences, Nanyang Technological University,Singapore 637616, Singapore
Abstract
In this paper, the homogeneous weights of matrix product codes overfinite principal ideal rings are studied and a lower bound for the minimumhomogeneous weights of such matrix product codes is obtained.
Key words:
Finite principal ideal ring, homogeneous weight, matrixproduct code.
Matrix product codes over finite fields were introduced in [1]. Many well-known constructions can be formulated as matrix product codes, for example,the ( a | a + b )-construction, the ( a + x | b + x | a + b + x )-construction, and, somequasi-cyclic codes can be rewritten as matrix product codes, see [13]. The refer-ence [1] also introduced non-singular by columns matrices and exhibited a lowerbound for the minimum Hamming distances of matrix product codes over finitefields constructed by such matrices. More references on matrix product codesappeared later, e.g., in [9, 10, 11, 15, 16].Codes over finite rings have also been studied from many perspectives sincethe seminal work [8]. It was also shown later in [18] that a finite Frobeniusring is suitable as an alphabet for linear coding. Further, [6] showed that, forany finite ring, there is a Frobenius module which is suitable as an alphabetfor linear coding. Inspired by the idea of module coding, [19] proved that thebiggest class of finite rings which are suitable as alphabets for linear codingconsists of the finite Frobenius rings.Finite principal ideal rings form an important subclass of finite Frobeniusrings. In particular, all the residue rings Z N of integers modulo an integer N >
In this paper, R is always a finite commutative ring.For the finite commutative ring R and a positive integer ℓ , any non-emptysubset C of R ℓ is called a code over R of length ℓ , or more precisely, an ( ℓ, M )code over R , where M = | C | denotes the cardinality of C ; the code C over R is said to be linear if C is an R -submodule of R ℓ . Recall that the usualHamming weight w H on R , i.e., w H (0) = 0 and w H ( r ) = 1 for all non-zero r ∈ R , induces in a standard way the Hamming weight on R ℓ , denoted by w H again, and the Hamming distance d H on R ℓ as follows: w H ( x ) = ℓ P i =1 w H ( x i ) for x = ( x , · · · , x ℓ ) ∈ R ℓ , and d H ( x , x ′ ) = w H ( x − x ′ ) for x , x ′ ∈ R ℓ . We also let d H ( C ) = min c = c ′ ∈ C d H ( c , c ′ ). This is known as the minimum Hamming distance of the code C .On the other hand, a homogeneous weight on the finite commutative ring R is defined to be a non-negative real function w h from R to the real number fieldwhich satisfies the following two conditions: • w h ( r ) = w h ( r ′ ) for r, r ′ ∈ R , provided Rr = Rr ′ , • there is a positive real number λ such that P x ∈ Rr w h ( x ) = λ | Rr | for anynon-zero r ∈ R , where | Rr | denotes the cardinality of the set Rr .It has been shown in [7] that such a function is uniquely determined (up to ascalar λ ) on R as follows w h ( r ) = λ (cid:18) − µ (0 , Rr ) ϕ ( Rr ) (cid:19) , (2.1)where µ is the M¨obius function on the lattice of all the principal ideals of R , and ϕ ( Rr ) denotes the number of elements x ∈ Rr such that Rx = Rr . Thus, the2omogeneous weight is uniquely determined up to a positive multiple λ . In therest of the paper, we always take λ = 1 in (2.1) for convenience, and denote theuniquely determined homogeneous weight by w h . As with the Hamming weight,the function w h on the ring R induces a function w h on R ℓ and a two-variablefunction d h on R ℓ ; and d h ( C ) = min c = c ′ ∈ C d h ( c , c ′ ) is said to be the minimumhomogeneous distance of the code C .Let A = ( a ij ) m × ℓ be an m × ℓ matrix over the finite commutative ring R ,and let C , · · · , C m be codes over R of length n . Then C = [ C , · · · , C m ] A = (cid:8) ( c , · · · , c m ) A | c ∈ C , · · · , c m ∈ C m (cid:9) is called a matrix product code , where the codewords c j of C j are written ascolumn vectors, hence ( c , · · · , c m ) are n × m matrices.We say that a square matrix over R is non-singular if its determinant is aunit of R . By usual linear algebra, a non-singular matrix over R is an invertiblematrix over R . Following [1], we say that the m × ℓ matrix A is non-singularby columns if, for any k ≤ m , any k × k determinant within the first k rowsof A is a unit of R . It is clear that, if A is non-singular by columns, thenany matrix obtained from A by permuting its columns is still non-singular bycolumns. We say that a matrix A is column-permutably upper triangular if A can be transformed by some suitable permutation of the columns to an uppertriangular matrix A ′ = ( a ′ ij ) m × ℓ (i.e., a ′ ij = 0 for all 1 ≤ j < i ≤ m ).From now on, we always assume that R is a finite commutative principal idealring , i.e., R is a finite commutative ring in which any ideal can be generatedby one element, or equivalently, there are finite chain rings R , · · · , R s and anisomorphism: R ∼ = −→ R × · · · × R s , r ( r , · · · , r s ) . (2.2)With this isomorphism, we can identify R with R × · · · × R s and write r =( r , · · · , r s ). For t = 1 , · · · , s , by J t we denote the unique maximal ideal of thechain ring R t (note that J t = 0 if R t is a field). Hence R t /J t is a finite field,and we further assume that F t = R t /J t ∼ = GF( q t ) for t = 1 , · · · , s, and q ≤ q ≤ · · · ≤ q s , (2.3)where GF( q t ) denotes the Galois field of order q t . For each t , there is an integer e t ≥
1, called the nilpotency index of the chain ring R t , such that J e t − t = 0 but J e t t = 0 , t = 1 , · · · , s. (2.4)Note that J t = R t for any t .We list some easy facts for later use. Since any ideal I of R has the form I = I × · · · × I s with I t being an ideal of R t for t = 1 , · · · , s , it follows that R/I = R /I × · · · × R s /I s is still a principal ideal ring.3ext, if elements u t , u t , · · · , u tq t of R t satisfy that F t = R t /J t = (cid:8) u t + J t , u t + J t , · · · , u tq t + J t (cid:9) , then, for any integer k with 0 < k ≤ e t and any element a of the set difference J k − t \ J kt , we have that J k − t /J kt = (cid:8) u t a + J kt , u t a + J kt , · · · , u tq t a + J kt (cid:9) . (2.5)Recall Formula (2.1) and rewrite it as (recall that we have set λ = 1): w h ( r ) = 1 − µ ( r ) ϕ ( r ) , where µ ( r ) = µ (0 , Rr ) and ϕ ( r ) = ϕ ( Rr ). Since R is a product of chain ringsas in Eqn (2.2) and ( r , · · · , r s ) ∈ R × · · · × R s , both µ and ϕ satisfy that µ (cid:0) ( r , · · · , r s ) (cid:1) = µ ( r ) · · · µ ( r s ) , ϕ (cid:0) ( r , · · · , r s ) (cid:1) = ϕ ( r ) · · · ϕ ( r s );thus, the homogeneous weight w h on R is (see [5, Theorem 4.1]) w h ( r ) = w h ( r , · · · , r s ) = 1 − Q st =1 µ ( r t ) ϕ ( r t ) . Further, for r t ∈ R t , there is (as long as r t = 0) a unique integer f r t with0 < f r t ≤ e t such that r t ∈ J e t − f rt t \ J e t − f rt +1 t , then we have (see [5] fordetails): µ ( r t ) = , r t = 0 , − , f r t = 1 , , f r t > , ϕ ( r t ) = , r t = 0 ,q t − , f r t = 1 ,q f rt t − q f rt − t , f r t > . For a non-zero element r = ( r , · · · , r s ) of R , setting T r = { ≤ t ≤ s | r t = 0 } , ¯ T r = { t ∈ T r | r t ∈ J e t − t } , (2.6)we obtain a formula to calculate the homogeneous weight on R as follows: w h ( r , · · · , r s ) = , r = 0 , , ¯ T r = T r , − ( − | T r | Q t ∈ T r q t − , ¯ T r = T r . (2.7)From Formula (2.7) one can see that (take any q ≥ q if s = 1):1 − q − q − ≤ w h ( r ) ≤ q − , ∀ = r ∈ R. (2.8)We next recall a few facts on matrices over a finite commutative principalideal ring R . For any t , by (2.3), we have a surjective homomorphism ρ t : R −→ F t , r ρ t ( r ) (2.9)with kernel I t = R × · · · × R t − × J t × R t +1 × · · · × R s , i.e., R/I t ∼ = R t /J t = F t .By a fundamental argument on determinants in linear algebra, one can prove(alternatively, a proof may be found in classical references such as [14]):4 emma 2.1. Let A = ( a ij ) m × ℓ be an m × ℓ matrix over R .(i) If A is non-singular by columns, then, for any non-trivial quotient ring ¯ R = R/I (i.e., I is an ideal of R with I = R ), the matrix ¯ A = (cid:0) ¯ a ij (cid:1) m × ℓ over ¯ R is non-singular by columns.(ii) If the matrix ρ t ( A ) = (cid:0) ρ t ( a ij ) (cid:1) m × ℓ over F t is non-singular by columns forall t = 1 , · · · , s , then A is non-singular by columns. By the above lemma and with the help of Eqn (2.5), it is easy to prove thefollowing result which is an extension of [1, Prop. 3.3] and [17, Prop. 1].
Lemma 2.2.
Assume that m > . There exists an m × ℓ matrix over R whichis non-singular by columns if and only if m ≤ ℓ ≤ min { q , · · · , q s } . The following result has appeared in [3, Lemma 4.1].
Lemma 2.3.
Assume that an m × ℓ matrix A over R is non-singular by columnsand ≤ k ≤ m . Then the minimum Hamming distance of the linear code in R ℓ generated by the first k rows of A is ℓ − k + 1 . We keep the notations of (2.2), (2.3) and (2.4). In this section, we state ourmain theorem, its corollaries and some remarks. The main theorem will beproved in the next section.
Theorem 3.1.
Let the notations be as in (2.2) and (2.3), and assume that q > q + 1 provided s > . Let A = ( a ij ) m × ℓ be an m × ℓ matrix over R which is non-singular by columns, and let C j be an ( n, M j ) code over R , for j = 1 , · · · , m . Then C = [ C , · · · , C m ] A is an (cid:0) nℓ, m Q j =1 M j (cid:1) -code over R with d h ( C ) ≥ min (cid:8) ℓd h ( C ) , ( ℓ − d h ( C ) , · · · , ( ℓ − m + 1) d h ( C m ) (cid:9) . (3.1) Furthermore, equality holds in (3.1) if one of the following conditions is satisfied: (C1) A is column-permutably upper triangular; (C2) C , C , · · · , C m are linear codes and C ⊇ C ⊇ · · · ⊇ C m . With the help of the results in [3], we have a consequence of the theorem forthe dual codes of matrix product codes.
Corollary 3.2.
Keep the notations as in (2.2), (2.3), and assume that q >q + 1 provided s > . If A is an m × m matrix over R which is non-singular by columns, and C j is an ( n, M j ) -linear code over R , for j = 1 , · · · , m , hen the dual code C ⊥ of the matrix product code C = [ C , · · · , C m ] A is an (cid:0) nm, m Q j =1 ( | R | n /M j ) (cid:1) -linear code over R with d h ( C ⊥ ) ≥ min (cid:8) md h ( C ⊥ m ) , ( m − d h ( C ⊥ m − ) , · · · , · d h ( C ⊥ ) (cid:9) . Furthermore, equality holds if one of the following conditions is satisfied: (C1) A is column-permutably upper triangular; (C2) C , C , · · · , C m are linear codes and C ⊇ C ⊇ · · · ⊇ C m . Proof.
For a square matrix A over R which is non-singular by columns, it isshown in [3, Theorem 3.3] that A is invertible and J ( A − ) T is non-singular bycolumns too, where ( A − ) T denotes the transpose of the inverse A − and J = · · · · · · · · · , and C ⊥ = [ C ⊥ , · · · , C ⊥ m ]( A − ) T . Noting that JJ = I , where I denotes theidentity matrix, and [ C ⊥ , · · · , C ⊥ m ] J = [ C ⊥ m , · · · , C ⊥ ], we have that C ⊥ = [ C ⊥ , · · · , C ⊥ m ] JJ ( A − ) T = [ C ⊥ m , · · · , C ⊥ ] J ( A − ) T . It is easy to check that, if A satisfies (C1) then so does J ( A − ) T ; and similarlyfor (C2). Thus the conclusions are derived from Theorem 3.1.In fact, in [3], a very precise description for the structure of C ⊥ , where C = [ C , · · · , C m ] A , was obtained in a more general setting, where R is anyfinite commutative Frobenius ring and A does not need to be square and non-singular by columns. It is therefore possible to obtain a lower bound for theminimum homogeneous distance of C ⊥ in that general setting, see [4]. Remark 3.3.
In the case when s = 1, i.e., R is a finite chain ring, Theorem 3.1contains the result [17, Proposition 2] and a generalization of the main resultof [9]; moreover, Corollary 3.2 bounds from below the homogeneous distance ofthe dual codes of matrix product codes.The residue ring Z N of integers modulo an integer N > N = p e · · · p e s s , where p < · · · < p s are primes and e t > t = 1 , · · · , s , we see that the Chinese RemainderTheorem: Z N ∼ = Z p e × · · · × Z p ess ,r (mod N ) (cid:0) r (mod p e ) , · · · , r (mod p e s s ) (cid:1) , is just the version for Z N of the decomposition (2.2). Therefore, the assumption“ q > q + 1 provided s >
1” in Theorem 3.1 translates into the assumption“ p = 3 provided p = 2” for Z N ; and we obtain the following result fromTheorem 3.1 at once. 6 orollary 3.4. Let
N > be an integer which is not divisible by . Let A be an m × ℓ matrix over Z N which is non-singular by columns, and let C j bean ( n, M j ) -code over Z N , for j = 1 , · · · , m . Then C = [ C , · · · , C m ] A is an (cid:0) nℓ, m Q j =1 M j (cid:1) -code over Z N with d h ( C ) ≥ min (cid:8) ℓd h ( C ) , ( ℓ − d h ( C ) , · · · , ( ℓ − m + 1) d h ( C m ) (cid:9) . (3.2) Furthermore, equality holds if one of the following conditions is satisfied: (C1) A is column-permutably upper triangular; (C2) C , C , · · · , C m are linear codes and C ⊇ C ⊇ · · · ⊇ C m . There is also an analogous version of Corollary 3.2 for Z N , with the sameassumption “ N is not divisible by 6”. Remark 3.5.
Recall that, to be a geometric distance, a two-variable real func-tion must meet three conditions: it is positive, it is symmetric, and it satisfiesthe triangle inequality. It is known that the homogeneous distance d h may notbe a geometric distance. References [2] and [12] contain extensive studies onweights on integral residue rings: in particular, a necessary and sufficient condi-tion for the homogeneous distance d h on Z ℓN to be a geometric distance is that N is not divisible by 6. By Corollary 3.4 and Example 3.6 below, this conditionis also necessary and sufficient for Inequality (3.2) to hold.The assumption “ q > q + 1 provided s >
1” in Theorem 3.1 will playa crucial role in the proof of the theorem. Moreover, the following exampleillustrates that the assumption cannot be removed.
Example 3.6.
Let the notations be as in (2.2), (2.3) and (2.4). Assume that s > q + 1 = q ≤ q ≤ · · · ≤ q s and let q = q . Let u t , u t , · · · , u tq t ∈ R t be as in Eqn (2.5). In the present case, we can choose them as follows: • for t = 1, u + J , · · · , u q + J are just all the elements of F = R /J ; • for t = 2, since q = q + 1, we can take u + J , · · · , u q + J to be allnon-zero elements of F = R /J ; • for t ≥
3, since q t > q , we can take u t + J t , · · · , u tq + J t to be distinctelements of F t = R t /J t .Let β j = ( u j , u j , · · · , u sj ) ∈ R = R × · · · × R s for j = 1 , · · · , q , and let A = (cid:18) · · · β β · · · β q (cid:19) . It is easy to see (cf. Lemma 2.2) that A is non-singular by columns. Let a = ( a , , , · · · , , b = (0 , b , , · · · , ∈ R = R × · · · × R s , a ∈ J e − \ { } and b ∈ J e − \ { } . Set C = Ra and C = Rb , thenboth are linear codes over R of length 1. Then we have the matrix product code C = [ C , C ] A . By Formula (2.7), we have that d h ( C ) = w h ( a ) = 1 + q − , d h ( C ) = w h ( b ) = 1 + q − = 1 + q . Since 1 + q − > q , we getmin { qd h ( C ) , ( q − d h ( C ) } = ( q − (cid:0) q (cid:1) = q − q . On the other hand, there is a codeword c of C as follows: c = ( a, b ) A = ( a + bβ , a + bβ , · · · , a + bβ q )with a + bβ j = ( a , b u j , , · · · , , j = 1 , · · · , q. By Eqn (2.5), b u j , for j = 1 , · · · , q , are all the non-zero elements of J e − ,and by Formula (2.7), w h ( a + bβ j ) = 1 − q − q − = 1 − q ( q − , so w h ( c ) = q (cid:0) − q ( q − (cid:1) = q − q − < q − q = min { qd h ( C ) , ( q − d h ( C ) } . Therefore, d h ( C ) < min { qd h ( C ) , ( q − d h ( C ) } , which implies that Inequality (3.1) does not hold for the matrix product code C = [ C , C ] A . We continue to keep the notations of (2.2), (2.3) and (2.4) and assume that q = q ≤ q ≤ · · · ≤ q s .Let A = ( a ij ) m × ℓ be a matrix over R which is non-singular by columns;then ℓ ≤ q = q if m > C , · · · , C m be codes over R oflength n . Consider the matrix product code C = [ C , · · · , C m ] A = { ( c , · · · , c m ) A | c ∈ C , · · · , c m ∈ C m } . (4.1)Since the proof of Inequality (3.1) is long and delicate, we put the key steps inSubsections 4.1–4.3; these subsections show that the following inequality holdsfor all 1 ≤ k ≤ m by splitting into various cases: w h (cid:0) ( c , · · · , c k , , · · · , ) A (cid:1) ≥ ( ℓ − k + 1) w h ( c k ) . Subsection 4.4 then completes the proof of Theorem 3.1.8 .1 k × ℓ Non-singular by Columns Matrices
Let A be as above, let 2 ≤ k ≤ m , and let A , · · · , A k be the first k rows of A .Let r , · · · , r k ∈ R with r k = 0 and let α = r A + · · · + r k A k ∈ R ℓ . Wehave seen from Lemma 2.3 that w H ( α ) = w H ( r A + · · · + r k A k ) ≥ ℓ − k + 1 . Lemma 4.1.
Let the notations be as above.(i) If w H ( r A + · · · + r k A k ) = ℓ − k + 1 , then w h ( r A + · · · + r k A k ) =( ℓ − k + 1) w h ( r k ) .(ii) If w H ( r A + · · · + r k A k ) > ℓ − k +1 and one of the following two conditionsholds: • k ≥ , • k = 2 and ℓ < q ,then w h ( r A + · · · + r k A k ) ≥ ( ℓ − k + 1) w h ( r k ) . Proof.
Write α = r A + · · · + r k A k = ( α , · · · , α ℓ ), where α j = r a j + · · · + r k a kj , j = 1 , · · · , ℓ. (i) Since w H ( r A + · · · + r k A k ) = ℓ − k +1, there are exact k − α , · · · , α ℓ . Without loss of generality, we assume that α = · · · = α k − = 0and α j = 0, for j = k, k + 1 , · · · , ℓ . In particular, r a + · · · + r k − a k − , = − r k a k r a + · · · + r k − a k − , = − r k a k ... ... ... ... r a ,k − + · · · + r k − a k − ,k − = − r k a k,k − . By the non-singularity by columns of A , the coefficient matrix ( a ji ) ( k − × ( k − of the above linear system is non-singular, i.e., invertible, hence r , · · · , r k − are all linear combinations of r k a k , · · · , r k a k,k − . Therefore, r j ∈ Rr k for j = 1 , · · · , k − , k ; hence all α j ∈ Rr k , i.e.,0 = Rα j ⊆ Rr k , j = k, k + 1 , · · · , ℓ. Suppose that there is an index t with k ≤ t ≤ ℓ such that Rα t $ Rr k . Con-sidering the quotient ring ¯ R = R/Rα t , then ¯ r k = 0 and ¯ A = (¯ a ij ) m × ℓ is stillnon-singular by columns (see Lemma 2.1). However,¯ α = · · · = ¯ α k − = 0 and ¯ α t = 0 , hence w H (¯ r ¯ A + · · · + ¯ r k ¯ A k ) ≤ ℓ − k < ℓ − k + 1 , w H (¯ r ¯ A + · · · + ¯ r k ¯ A k ) ≥ ℓ − k + 1 (see Lemma2.3). Therefore, Rα j = Rr k for all j = k, k + 1 , · · · , ℓ , and w h ( r A + · · · + r k A k ) = ℓ X j = k w h ( α j ) = ℓ X j = k w h ( r k ) = ( ℓ − k + 1) w h ( r k ) . (ii) In this case, there are at least ℓ − k + 2 non-zeros among α , · · · , α ℓ . ByInequality (2.8), we get w h ( α ) ≥ ( ℓ − k + 2) (cid:16) − q − q − (cid:17) (4.2)and ( ℓ − k + 1) (cid:16) q − (cid:17) ≥ ( ℓ − k + 1) w h ( r k ) . (4.3)It is an elementary calculation to check that( x + 1) (cid:16) − q − q − (cid:17) ≥ x (cid:16) q − (cid:17) ⇐⇒ x ≤ q − − q q . (4.4)Recall that ℓ ≤ q ≤ q . If k ≥
3, or if k = 2 and ℓ < q , then ℓ − k + 1 ≤ q − − q q . In both cases, we can apply Formula (4.4) to (4.2) and (4.3), with x = ℓ − k + 1, and obtain w h ( α ) ≥ ( ℓ − k + 1) w h ( r k ). Proposition 4.2.
Let A and c ∈ C , · · · , c k ∈ C k be as in (4.1) and assumethat c k = . If one of the following two conditions holds: • k ≥ , • k = 2 and ℓ < q ,then w h (cid:0) ( c , · · · , c k , , · · · , ) A (cid:1) ≥ ( ℓ − k + 1) w h ( c k ) . Proof.
Let c i k , · · · , c i w k be all the non-zero entries of c k = ( c k , · · · , c nk ) T .Then w h ( c k ) = w h ( c i k ) + · · · + w h ( c i w k ). Noting that the i th row of the matrix( c , · · · , c k , , · · · , ) A is c i A + · · · + c ik A k , where A , · · · , A k are as above,we have w h (cid:0) ( c , · · · , c k , , · · · , ) A (cid:1) = P ni =1 w h (cid:0) c i A + · · · + c ik A k (cid:1) ≥ P wt =1 w h (cid:0) c i t A + · · · + c i t k A k (cid:1) ≥ P wt =1 ( ℓ − k + 1) w h ( c i t k )= ( ℓ − k + 1) w h ( c k ) , where the second “ ≥ ” follows from Lemma 4.1.10 .2 × q Non-singular by Columns Matrices
In the following, we let q = q and assume that A = (cid:18) · · · β β · · · β q (cid:19) is a2 × q matrix over R which is non-singular by columns. Write β j = ( u j , u j , · · · , u sj ) ∈ R × R × · · · × R s . Let a, b ∈ R with b = 0, and write a = ( a , · · · , a s ) and b = ( b , · · · , b s ) with a t , b t ∈ R t for t = 1 , · · · , s . Consider the word α = ( a, b ) A = ( α , · · · , α q ) ∈ R q , (4.5)where α j = a + bβ j = (cid:0) a + b u j , · · · , a s + b s u sj (cid:1) , j = 1 , · · · , q. Then w H ( α ) ≥ q −
1. From Lemma 4.1(i), we have seen that w h ( α ) = ( q − w h ( b ) if w H ( α ) = q − . (4.6)In the following, we further assume that α j = 0 for all j = 1 , · · · , q . Lemma 4.3. If w h ( b ) = 1 , then w h ( α ) ≥ ( q − w h ( b ) . Proof.
Since w h ( b ) = 1, there is at least one k such that b k / ∈ J e k − k . Take I k = R × · · · × R k − × J e k − k × R k +1 × · · · × R s , and consider the quotient ring¯ R k := R/I k ∼ = R k /J e k − k . Then the matrix ¯ A over ¯ R k is still non-singular bycolumns (see Lemma 2.1), ¯ b = ¯ b k = 0, and the elements α j = (cid:0) a + b u j , · · · , a k + b k u kj , · · · , a s + b s u sj (cid:1) , j = 1 , · · · , q, are mapped to ¯ α j = ¯ a k + ¯ b k ¯ u kj , j = 1 , · · · , q. Then, for the word ¯ α = (¯ a, ¯ b ) ¯ A = (cid:0) ¯ α , · · · , ¯ α q (cid:1) over ¯ R k , its Hamming weightsatisfies w H (¯ α ) ≥ q −
1. Since q ≥
2, there is at least one non-zero entry, say¯ α t = 0, i.e., a k + b k u kt / ∈ J e k − k . Hence, w h ( α t ) = 1. Noting that w h ( α j ) ≥ − q − q − for j = t (see Formula (2.8)), we have w h ( α ) = q P j =1 w h ( α j ) ≥ q − (cid:16) − q − q − (cid:17) ≥ q − q − w h ( b ) . ✷ Note that w h ( b ) = 1 if and only if b t ∈ J e t − t for t = 1 , · · · , s . Lemma 4.4. If b = ( b , · · · , b s ) with b t ∈ J e t − t , for t = 1 , · · · , s , and w h ( a ) =1 , then w h ( α ) ≥ ( q − w h ( b ) . roof. Similar to the proof above, we can assume that a k / ∈ J e k − k for some k .Since b k ∈ J e k − k , it follows that a k + b k u kj / ∈ J e k − k for all j = 1 , · · · , q . Thus w h ( α j ) = 1 for all j = 1 , · · · , q , and w h ( α ) = q P j =1 w h ( α j ) = q = ( q − (cid:16) q − (cid:17) ≥ ( q − w h ( b ) . ✷ From now on, we further assume that a = ( a , · · · , a s ) , b = ( b , · · · , b s ) with a t , b t ∈ J e t − t for t = 1 , · · · , s, (4.7)and let T a = { ≤ t ≤ s | a t = 0 } , T b = { ≤ t ≤ s | b t = 0 } , T = T a ∪ T b . (4.8) Lemma 4.5.
Let t = min t ∈ T t . If q < q t , then w h ( α ) ≥ ( q − w h ( b ) . Proof.
Since b = (0 , · · · , , b t , · · · , b s ), by Formula (2.7), we have that w h ( b ) ≤ q t − . On the other hand, a j + b j u tj = 0 for any t < t , so α j = (0 , · · · , , a j + b j u t j , · · · , a j + b j u sj ) , hence w h ( α j ) ≥ − q t − q t − (when t = s , set q t +1 to be any integergreater than q t ). Since q − ≤ q t − ≤ q t − − q t q t , we can use (4.4) toobtain w h ( α ) = q P j =1 w h ( α j ) ≥ q (cid:16) − q t − q t − (cid:17) ≥ ( q − (cid:16) q t − (cid:17) ≥ ( q − w h ( b ) . ✷ In the following, we further assume that1 ∈ T , and q > q + 1 if s >
1. (4.9)
Lemma 4.6. If T b = { t ′ } contains only one index t ′ with ≤ t ′ ≤ s , then w h ( α ) ≥ ( q − w h ( b ) . Proof.
First, assume that t ′ = 1. Then b = ( b , , · · · , ∈ R × · · · × R s with0 = b ∈ J e − , so w h ( b ) = 1 + q − (recall that q = q ), and α j = (cid:0) a + b u j , a , · · · , a s (cid:1) . Taking I = J × R × · · · × R s and ¯ R = R/I ∼ = R /J = F , then¯ A = (cid:18) · · · u ¯ u · · · ¯ u q (cid:19)
12s a matrix over the field F which is still non-singular by columns, so as elementsof the field F , the entries ¯ u , ¯ u , · · · , ¯ u q must be distinct. Since | F | = q ,we conclude that ¯ u , ¯ u , · · · , ¯ u q must consist of all the elements of F . ByEqn (2.5), b u , b u , · · · , b u q are just all the elements of J e − , hence a + b u , a + b u , · · · , a + b u q are again just all the elements of J e − . In other words, exactly one of them is0, and the other ( q −
1) terms are non-zero. By Formula (2.7), w h ( α j ) = − ( − | T | · Q t ∈ T q t − , if a + b u j = 0,1 + ( − | T | · Q = t ∈ T q t − , if a + b u j = 0.Therefore, w h ( α ) = q X j =1 w h ( α j )= − | T | · Y = t ∈ T q t − + ( q − − ( − | T | · Y t ∈ T q t − ! = 1 + ( − | T | · Y = t ∈ T q t − q − − ( − | T | · Y = t ∈ T q t − q = ( q − (cid:18) q − (cid:19) = ( q − w h ( b ) . Note that the above argument still works well for T = { } (in particular, itworks well for s = 1) provided we adopt the convention that Q = t ∈ T q t − = 1.Next, we assume that t ′ >
1. Then s ≥ w h ( b ) ≤ q − ,w h ( α j ) ≥ − q − q − , j = 1 , · · · , q. Since q ≥ q + 2, i.e., q − ≥ q + 1, and q ≥
2, it follows that: w h ( α ) − ( q − w h ( b ) ≥ q (cid:16) − q − q − (cid:17) − ( q − (cid:16) q − (cid:17) = 1 − q ( q − q − − q − q − = 1 − q ( q − q − − ( q − ( q − q − ≥ − q +( q − ( q − q +1) = q − q − ≥ . In other words, w h ( α ) ≥ ( q − w h ( b ).13 emma 4.7. If | T b | ≥ , then w h ( α ) ≥ ( q − w h ( b ) . Proof.
By Formula (2.7), w h ( b ) = 1 − ( − | T b | Q t ∈ T b q t − , thus w h ( b ) ≤ q − q − q − q ≥ q if s = 2). On the other hand, by (2.8), we have w h ( α j ) ≥ − q − q − . Noting that q − ≥ q + 1 and q − > q − ≥
1, we have that w h ( α ) − ( q − w h ( b ) ≥ q (cid:16) − q − q − (cid:17) − ( q − (cid:16) q − q − q − (cid:17) = 1 − q ( q − q − − q − q − > − q ( q − q +1) − q +1)( q − = 1 − q − ≥ . We have obtained the desired inequality w h ( α ) ≥ ( q − w h ( b ).Summarizing Eqn (4.6) and Lemmas 4.3–4.7, we have that, if q > q + 1provided s >
1, then the homogeneous weight of the word (4.5) satisfies w h (cid:0) ( a, b ) A (cid:1) ≥ ( q − w h ( b ) . (4.10)Thus, similar to Proposition 4.2, we obtain the following conclusion. Proposition 4.8.
Let A = ( a ij ) m × q be non-singular by columns, and let c ∈ C , c ∈ C and c = . Assume the following condition holds • q > q + 1 provided s > .Then w h (cid:0) ( c , c , , · · · , ) A (cid:1) ≥ ( q − w h ( c ) . Proof.
It is clear that, for any q × q diagonal matrix D whose diagonalentries are all units of R , we have that w h (cid:0) ( c , c , , · · · , ) AD (cid:1) = w h (cid:0) ( c , c , , · · · , ) A (cid:1) . (4.11)Since A is non-singular by columns, any element of the first row of A is a unitof R , so there is a suitable diagonal matrix D such that all entries of the firstrow of AD are 1. Thus, we can assume that the first row of A is the all-1 vector.Then, as in the proof of Proposition 4.2, we can obtain the conclusion of theproposition by using (4.10) . 14 .3 × ℓ Non-singular by Columns Matrices
A 1 × ℓ non-singular by columns matrix is none other than a matrix consisting ofonly one row, all of whose entries are units. This is essentially the key ingredientin the proof of the following result. Proposition 4.9.
Let A = ( a ij ) m × ℓ be non-singular by columns, and = c =( c , · · · , c n ) T ∈ C . Then w h (cid:0) ( c , , · · · , ) A (cid:1) ≥ ℓw h ( c ) . Proof.
By the non-singularity by columns of A , all the entries a , · · · , a ℓ ofthe first row of A are units in R . Thus, w h (cid:0) ( c , , · · · , ) A (cid:1) = ℓ X j =1 w h ( a j c ) = ℓ X j =1 w h ( c ) = ℓw h ( c ) . ✷ Now we can complete the proof of Theorem 3.1.First, we prove Inequality (3.1).Let c = ( c , · · · , c m ) A and c ′ = ( c ′ , · · · , c ′ m ) A be any two distinct code-words of the code C . Then, not all of b j = c j − c ′ j , for j = 1 , · · · , m , are zero.Hence, c − c ′ = ( b , · · · , b m ) A = and d h ( c , c ′ ) = w h ( c − c ′ ) = w h (cid:0) ( b , · · · , b m ) A (cid:1) . It is enough to show that d h ( c , c ′ ) is bounded below by one of the entries inthe braces of the right hand side of (3.1) of Theorem 3.1. Since not all of b , · · · , b m are , there is an index k with 1 ≤ k ≤ m such that b k = but b k +1 = · · · = b m = .If k = 1, by Proposition 4.9, we have d h ( c , c ′ ) = w h (cid:0) ( b , , · · · , ) A (cid:1) ≥ ℓw h ( b ) = ℓw h ( c − c ′ ) ≥ ℓd h ( C ) . Suppose that k = 2, then m ≥ ℓ ≤ q .If ℓ < q , by Proposition 4.2, we have d h ( c , c ′ ) = w h (cid:0) ( b , b , , · · · , ) A (cid:1) ≥ ( ℓ − w h ( b )= ( ℓ − w h ( c − c ′ ) ≥ ( ℓ − d h ( C ) . Otherwise, ℓ = q , and by Proposition 4.8, we still have d h ( c , c ′ ) = w h (cid:0) ( b , b , , · · · , ) A (cid:1) ≥ ( ℓ − w h ( b ) ≥ ( ℓ − d h ( C ) . The remaining case is that of k >
2. By Proposition 4.2, we obtain d h ( c , c ′ ) = w h (cid:0) ( b , · · · , b k , , · · · , ) A (cid:1) ≥ ( ℓ − k + 1) w h ( b k )= ( ℓ − k + 1) w h ( c k − c ′ k ) ≥ ( ℓ − k + 1) d h ( C k ) . A is column-permutably upper triangular. Since anypermutation of columns does not change the weights and other parameters ofthe resulting codewords, we can assume that A is upper triangular: A = a a · · · a m · · · a ℓ a · · · a m · · · a ℓ . . . ... ... ... a mm · · · a mℓ . Since A is non-singular by columns, every element of the first row is a unitof R . Similarly, every (2 × (cid:18) a a j a j (cid:19) is a unit, i.e., every a j , for j = 2 , · · · , ℓ , is a unitof R . Continuing this reasoning, we see that • all a ij for 1 ≤ i ≤ m and i ≤ j ≤ ℓ are units of R .For any k with 1 ≤ k ≤ m , take c k , c ′ k ∈ C k such that d h ( c k , c ′ k ) = d h ( C k ). Wehave two codewords of C as follows: c = ( , · · · , , c k , , · · · , ) A, c ′ = ( , · · · , , c ′ k , , · · · , ) A, whose homogeneous distance is d h ( c , c ′ ) = w h ( c − c ′ ) = w h (cid:0) ( , · · · , , c k − c ′ k , , · · · , ) A (cid:1) = w h (cid:0) , · · · , , a kk ( c k − c ′ k ) , · · · , a kℓ ( c k − c ′ k ) (cid:1) = ℓ X j = k w h (cid:0) a kj ( c k − c ′ k ) (cid:1) = ℓ X j = k w h ( c k − c ′ k )= ( ℓ − k + 1) d h ( C k ) . Thus d h ( C ) ≤ min { ℓd h ( C ) , · · · , ( ℓ − m + 1) d h ( C m ) } . It follows that equalitymust hold in (3.1).Finally, assume that C , · · · , C m are linear and C ⊇ · · · ⊇ C m . Write A = ( a ij ) m × ℓ . Since a is a unit of R , we can add a suitable multiple of thefirst row to the i th row, for each 2 ≤ i ≤ m , such that the entries of the firstcolumn of A below a are changed into 0, that is, there are b , · · · , b m ∈ R such that b b m a a · · · a ℓ a a · · · a ℓ ... ... ... ... a m a m · · · a mℓ = a a · · · a ℓ a ′ · · · a ′ ℓ ... ... ... a ′ m · · · a ′ mℓ . Similarly, a ′ is also a unit of R , and we can add a suitable multiple of thesecond row to the i th row, for 3 ≤ i ≤ m , such that the entries below a ′ of the16econd column are changed into 0. Continuing in the same manner, we obtaina lower triangular m × m matrix P = b b m b m · · · such that P A is an upper triangular matrix, which is still non-singular bycolumns.Since • C = [ C , · · · , C m ] A = (cid:0) [ C , · · · , C m ] P − (cid:1) ( P A ), • P − still has the form P − = b ′ b ′ m b ′ m · · · , • [ C , · · · , C m ] P − = [ C , · · · , C m ] (since C , · · · , C m are linear and C ⊇ C ⊇ · · · ⊇ C m ),it follows that C = [ C , · · · , C m ]( P A ) , where P A is upper triangular. Hence, by the result above, equality holdsin (3.1).
Acknowledgements
Quite a part of this work was done while the first and third authors were visitingthe Division of Mathematical Sciences, School of Physical and MathematicalSciences, Nanyang Technological University, Singapore, in Autumn 2011. Theyare grateful for the hospitality and support. They also thank NSFC for thesupport through Grants No. 11271005 and No. 11171370. Part of the workwas also done when the second author was visiting the School of Mathematicsand Statistics, Central China Normal University, Wuhan, in Spring 2012. Theauthor acknowledges the support and hospitality received. The work of thisauthor was partially supported by Singapore National Research FoundationCompetitive Research Programme NRF-CRP2-2007-03.
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