On the Weight Hierarchy of Locally Repairable Codes
aa r X i v : . [ c s . I T ] F e b On the Weight Hierarchy of LocallyRepairable Codes
Jie Hao ∗ , Shu-Tao Xia ∗ , Bin Chen ∗† , and Fang-Wei Fu ‡∗ Graduate School at Shenzhen, Tsinghua University, Shenzhen, China † School of Mathematical Sciences, South China Normal University, Guangzhou, China ‡ Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, ChinaEmail: [email protected], [email protected], [email protected], [email protected]
Abstract An ( n, k, r ) locally repairable code (LRC) is an [ n, k, d ] linear code where every code symbol can be repaired from at most r other code symbols. An LRC is said to be optimal if the minimum distance attains the Singleton-like bound d ≤ n − k −⌈ k/r ⌉ +2 .The generalized Hamming weights (GHWs) of linear codes are fundamental parameters which have many useful applications.Generally it is difficult to determine the GHWs of linear codes. In this paper, we study the GHWs of LRCs. Firstly, we obtain ageneralized Singleton-like bound on the i -th (1 ≤ i ≤ k ) GHWs of general ( n, k, r ) LRCs. Then, it is shown that for an optimal ( n, k, r ) LRC with r | k , its weight hierarchy can be completely determined, and the i -th GHW of an optimal ( n, k, r ) LRC with r | k attains the proposed generalized Singleton-like bound for all ≤ i ≤ k . For an optimal ( n, k, r ) LRC with r ∤ k , we givelower bounds on the GHWs of the LRC and its dual code. Finally, two general bounds on linear codes in terms of the GHWs arepresented. Moreover, it is also shown that some previous results on the bounds of minimum distances of linear codes can also beexplained or refined in terms of the GHWs. I. I
NTRODUCTION
Locally repairable codes have attracted a lot of interest recently. An ( n, k, r ) LRC is an [ n, k, d ] linear code with localityconstraint on the code symbols, i.e., each of the n code symbols can be repaired by accessing at most r other code symbols.The minimum distance of an LRC satisfies the well-known Singleton-like bound [1] d ≤ n − k − (cid:24) kr (cid:25) + 2 . (1)When r = k , the above bound reduces to the classical Singleton bound d ≤ n − k + 1 [2]. Further studies on the bounds ofLRCs can be found in [3]-[5]. Many works have proposed optimal LRCs attaining the Singleton-like bound (1), e.g., [5]-[9],among which codes with small field size are of particular interest. The elegant Reed-Solomon-like optimal LRCs proposed byTamo and Barg in [5] require the field size to be just slightly greater than the code length. The codes were further generalizedto optimal cyclic LRCs in [7] and LRCs on algebraic curves [8]. All the possible four classes of optimal binary LRCs attainingthe bound (1) were found in [9]. A bound of linear codes with a local-error-correction property were proposed in [10] whichcould include the bound (1) as a special case.The generalized Hamming weight (GHWs) [11][12] are fundamental parameters of linear codes and were first used by Weiin cryptography to fully characterize the performance of linear codes in a wire-tap channel of type II [11]. Let C be a q -ary [ n, k, d ] linear code and D be a subcode of C . Let dim( D ) denote the dimension of D . The support of a vector is the set ofcoordinates of its non-zero components. If a coordinate is in the support of a vector, it is said to be covered by the vector.The support of D is defined to be supp( D ) = { i : ∃ ( c , c , · · · , c n ) ∈ D , c i = 0 } . For ≤ i ≤ k , the i -th GHW of C is defined to be d i = min {| supp( D ) | : D ⊂ C and dim( D ) = i } , where | supp( D ) | denotes the cardinality of the set supp( D ) . Note that d is the minimum distance of C . The i -th (1 ≤ i ≤ k ) GHW of C satisfies the generalized Singleton bound d i ≤ n − k + i. (2)MDS codes meet the generalized Singleton bound for all ≤ i ≤ k . The weight hierarchy of C is the set of integers { d i | ≤ i ≤ k } . For an [ n, k ] linear code C , ≤ d < d < · · · < d k = n . The gap numbers of C is the complement of theset of its weight hierarchy, denoted as { g i , ≤ i ≤ n − k } = [ n ] \ { d i | ≤ i ≤ k } . Let C ⊥ be the dual code of C and itsweight hierarchy and gap numbers respectively be { d ⊥ i | ≤ i ≤ n − k } and { g ⊥ i | ≤ i ≤ k } . The weight hierarchy of C and C ⊥ satisfies the following property of duality. { d i | ≤ i ≤ k } = { , , · · · , n } \ { n + 1 − d ⊥ j | ≤ j ≤ n − k } . (3)n terms of the gap numbers of C ⊥ , d i = ( n + 1) − g ⊥ k − i +1 , ≤ i ≤ k. (4)The minimum distance d = d = ( n + 1) − g ⊥ k , where g ⊥ k = max { k + i : 1 ≤ i ≤ n − k, d ⊥ i < k + i } , (5) = min { k + i : 1 ≤ i ≤ n − k, d ⊥ i = k + i } − . (6)Many research works have been devoted to determine or estimate the GHWs of many series of linear codes, such as Hammingcodes [11], Reed-Muller codes [11], [13], BCH codes and their dual codes [14]-[16], etc.. Generally speaking, it is difficult todetermine the GHWs of linear codes, and the complete weight hierarchy is known for only a few cases. Recently some worksstarted to analyze the GHWs of particular LRCs. The GHWs of LRCs on algebraic curves were studied in [17]. Lalitha et al. [18] studied the GHWs and weight distributions of maximally recoverable codes, which are variants of optimal LRCs [19].In this paper, we study the GHWs of LRCs. Firstly, we obtain a generalized Singleton-like bound on the i -th ( ≤ i ≤ k )GHWs of general ( n, k, r ) LRCs. When i = 1 , the proposed generalized Singleton-like bound gives the Singleton-like bound(1). When r = k , i.e., there is no locality constraint, the proposed generalized Singleton-like bound reduces to the classicalgeneralized Singleton bound (2). Then, for optimal ( n, k, r ) LRCs attaining the Singleton-like bound (1), some lower boundson the GHWs of optimal LRCs and their dual codes are obtained. Surprisingly, it is shown that for an ( n, k, r ) optimal LRC C meeting the Singleton-like bound (1) with r | k , its weight hierarchy can be completely determined as d i = n − k − (cid:24) k − i + 1 r (cid:25) + i + 1 , ≤ i ≤ k. Its dual code C ⊥ has weight hierarchy d ⊥ i = (cid:26) i ( r + 1) , ≤ i ≤ k/r,k + i, k/r < i ≤ n − k. Finally, we employ a parity-check matrix approach [3] to give general upper bounds of linear codes in terms of the GHWsof their dual codes, which can include several known bounds of LRCs as special cases. It is also shown that some resultsproposed in [6][21] on bounds of minimum distances of linear codes can be explained or refined in terms of the GHWs.The rest of this paper is organized as follows. In Section II, upper bounds on the GHWs of general LRCs are presented.Section III studies the GHWs of optimal LRCs. Section IV analyzes the bounds of the linear codes in terms of the GHWs ofthe dual codes. Section V concludes the paper.II. U
PPER B OUNDS ON THE
GHW
S OF G ENERAL
LRC S In this section, we give some upper bounds on the GHWs of LRCs and their dual codes. Specially a generalized Singleton-like bound on the i -th ( ≤ i ≤ k ) GHWs of general ( n, k, r ) LRCs is obtianed. The proposed generalized Singleton-likebound can give the Singleton-like bound (1) when i = 1 and reduce to the classical generalized Singleton bound (2) whenthere is no locality constraint. Lemma 1.
Let C be an ( n, k, r ) LRC and C ⊥ be its dual code. The i -th ( ≤ i ≤ n − k ) GHW of the dual code C ⊥ satisfies d ⊥ i ≤ (cid:26) i ( r + 1) , ≤ i ≤ ⌊ k/r ⌋ ,k + i, ⌊ k/r ⌋ + 1 ≤ i ≤ n − k. (7) Proof.
Since every code symbol in an ( n, k, r ) LRC C has locality r , each coordinate is covered by at least one parity-checkequation with weight at most r + 1 from C ⊥ . By [3], ⌊ k/r ⌋ ≤ n − k . For ≤ i ≤ ⌊ k/r ⌋ , let us select a subcode D ⊥ of C ⊥ with dimension i . For the first coordinate, select a parity-check equation with weight at most r + 1 to cover it; then, for thefirst uncovered coordinate, select another parity-check equation with weight at most r + 1 to cover it; repeating the procedureiteratively until we obtain i parity-check equations. Apparently, these i parity-check equations are linearly independent, whichimplies that these i vectors are a basis of a subcode D ⊥ with dimension i . Since each of these i vectors has weight ≤ r + 1 , | supp( D ⊥ ) | ≤ i ( r +1) . The generalized Singleton bound says that d ⊥ i ≤ k + i . For ≤ i ≤ ⌊ k/r ⌋ , it holds that i ( r +1) ≤ k + i .Hence for ≤ i ≤ ⌊ k/r ⌋ , d ⊥ i ≤ | supp( D ⊥ ) | ≤ i ( r + 1) . For ⌊ k/r ⌋ + 1 ≤ i ≤ n − k , since i ( r + 1) > k + i , we have d ⊥ i ≤ k + i . Thus the conclusion follows. Lemma 2.
Let C be an ( n, k, r ) LRC and C ⊥ be its dual code. For ≤ i ≤ ⌊ k/r ⌋ , the GHWs of the dual code C ⊥ satisfies d ⊥ i +1 ≤ d ⊥ i + ( r + 1) . (8) Proof.
Suppose D ⊥ i is a subcode of C ⊥ with dimension i and | supp( D ⊥ i ) | = d ⊥ i . Let H i = [( h T , · · · h Ti ) T ] be a basis of D ⊥ i with i independent parity-check equations from D ⊥ i . Choose a coordinate j from the set [ n ] \ supp( D ⊥ i ) . Then select aarity-check equation ¯ h with weight at most r +1 to cover the coordinate j . Apparently ¯ h is independent with the vectors in H i ,which indicates that H i +1 = [ H Ti , ¯ h T ] T is the basis of a subcode D ⊥ i +1 of C ⊥ with dimension i + 1 . Since | supp(¯ h ) | ≤ r + 1 ,we have | supp( D ⊥ i +1 ) | ≤ d ⊥ i + ( r + 1) . Thus it follows that d ⊥ i +1 ≤ | supp( D ⊥ i +1 ) | ≤ d ⊥ i + ( r + 1) . Lemma 3.
Let C be an ( n, k, r ) LRC and C ⊥ be its dual code. If the i -th ( < i ≤ ⌊ k/r ⌋ ) GHW of C ⊥ satisfies that d ⊥ i = i ( r + 1) . Then for any ≤ j < i , the j -th GHW d ⊥ j = j ( r + 1) . Proof.
By Lemma 2, d ⊥ j ≥ d ⊥ j +1 − ( r + 1) , ≤ j ≤ ⌊ k/r ⌋ − . Combining that d ⊥ i = i ( r + 1) , we obtain that for any ≤ j < i , the j -th GHW d ⊥ j ≥ j ( r + 1) . According to Lemma 1, d ⊥ j ≤ j ( r + 1) . Thus the conclusion follows.
Lemma 4.
Let C be an ( n, k, r ) LRC and C ⊥ be its dual code. Then the gap numbers of the C ⊥ satisfy g ⊥ i ≥ (cid:24) ir (cid:25) + i − , ≤ i ≤ k. (9) Proof.
The lemma is proved by induction. For i = 1 , it follows that g ⊥ = 1 , since that if g ⊥ ≥ , then d ⊥ = 1 , which implies C contains a coordinate which is always zero for all the codewords of C . Suppose that for ≤ i ≤ k − , g ⊥ i ≥ (cid:6) ir (cid:7) + i − .Then consider the gap number g ⊥ i +1 ,1) If (cid:6) i +1 r (cid:7) = (cid:6) ir (cid:7) , by g ⊥ i +1 ≥ g ⊥ i + 1 , we have g ⊥ i +1 ≥ ( (cid:6) ir (cid:7) + i −
1) + 1 = (cid:6) i +1 r (cid:7) + ( i + 1) − .2) If (cid:6) i +1 r (cid:7) = (cid:6) ir (cid:7) + 1 , i.e., i = ar, i + 1 = ar + 1 . By Lemma 1, d ⊥ a ≤ a ( r + 1) . Hence the set { , · · · , a ( r + 1) } containsat least { d ⊥ , · · · , d ⊥ a } . In other words, the set { , · · · , a ( r + 1) } contains at most these gap numbers { g ⊥ · · · g ⊥ ar } . Thenit follows that g ⊥ ar +1 ≥ a ( r + 1) + 1 , i.e., g ⊥ i +1 ≥ (cid:6) i +1 r (cid:7) + i .Combining the above two cases, the conclusion follows. Theorem 1.
Let C be an ( n, k, r ) LRC, the i -th ( ≤ i ≤ k ) GHW of C satisfies the generalized Singleton-like bound d i ≤ n − k − (cid:24) k − i + 1 r (cid:25) + i + 1 . (10) Proof.
For ≤ i ≤ k , the i -th GHW satisfies d i ( a ) = n + 1 − g ⊥ k − i +1( b ) ≤ n + 1 − (cid:18)(cid:24) k − i + 1 r (cid:25) + k − i (cid:19) = n − k − (cid:24) k − i + 1 r (cid:25) + i + 1 , where ( a ) follows from (4) and ( b ) follows from Lemma 4. Remark 1.
Note that when i = 1 , the generalized Singleton-like bound (10) gives the Singleton-like bound (1). Remark 2.
When r = k , i.e., there is no locality constraint on the code symbols of C , the generalized Singleton-like bound(10) reduces to the classical generalized Singleton bound d i ≤ n − k + i . III. T HE GHW
S OF O PTIMAL
LRC S In this section, we will focus on the GHWs of optimal LRCs meeting the Singleton-like bound (1). It is shown that for anoptimal ( n, k, r ) LRC C with r | k , the weight hierarchy of C and its dual code C ⊥ can be completely determined. For theother case that r ∤ k , we give some lower bounds on the GHWs. . The Weight Hierarchy of Optimal LRCs with r | k Theorem 2.
Let C be an optimal ( n, k, r ) LRC meeting the Singleton-like bound (1) with r | k and C ⊥ be its dual code. Theweight hierarchy of C ⊥ can be completely determined as d ⊥ i = (cid:26) i ( r + 1) , ≤ i ≤ k/r,k + i, k/r + 1 ≤ i ≤ n − k. (11) Proof.
By the duality property (4), d ⊥ i = ( n + 1) − g n − k − i +1 , ≤ i ≤ n − k. (12)Since d = d = n − k − k/r + 2 , C has gap numbers g s = s , where ≤ s ≤ n − k − k/r + 1 . Hence by (4), it follows thatfor k/r ≤ i ≤ n − k , d ⊥ i = k + i. Since d ⊥ k/r = k + k/r = kr ( r + 1) , combining Lemma 3, it holds that for ≤ i ≤ k/r , d ⊥ i = i ( r + 1) . Hence the conclusion follows.Since the duality properties (3) and (4), we can obtain the following theorem.
Theorem 3.
Let C be an optimal ( n, k, r ) LRC meeting the Singleton-like bound (1) with r | k . The weight hierarchy of C can be completely determined as d i = n − k − (cid:24) k − i + 1 r (cid:25) + i + 1 , ≤ i ≤ k. (13) Proof.
By Theorem 2, it is not hard to verify that the gap numbers of the C ⊥ are g ⊥ i = i + (cid:24) ir (cid:25) − , ≤ i ≤ k. (14)Then by the duality property (4), it follows that for ≤ i ≤ k , d i = n + 1 − g ⊥ k − i +1 = n − k − (cid:24) k − i + 1 r (cid:25) + i + 1 , which completes the proof. Remark 3.
For an ( n, k, r ) optimal LRC meeting the Singleton-like bound (1) with r | k , the i -th ( ≤ i ≤ k ) GHW attainsthe generalized Singleton-like bound (10) with equality. Example 1.
For an ( n = 12 , k = 6 , r = 3) optimal cyclic LRC [7] C with r | k , by Theorem 2, the weight hierarchy of itsdual code C ⊥ can be completely determined as d ⊥ = 4 , d ⊥ = 8 , d ⊥ = 9 , d ⊥ = 10 , d ⊥ = 11 , d ⊥ = 12 . By Theorem 3, the complete weight hierarchy of C is d = 6 , d = 7 , d = 8 , d = 10 , d = 11 , d = 12 . B. Lower Bounds on the GHWs of Optimal LRCs
Lemma 5.
Let C be an optimal ( n, k, r ) LRC meeting the Singleton-like bound (1) and C ⊥ be its dual code. The i -th GHWof the dual code C ⊥ satisfies d ⊥ i ≥ i ( r + 1) − ⌈ k/r ⌉ r + k, ≤ i ≤ ⌈ k/r ⌉ − , (15) d ⊥ i = k + i, i ≥ ⌈ k/r ⌉ . (16) Proof.
Since the st GHW of C is d = n − k − ⌈ k/r ⌉ + 2 , g n − k −⌈ k/r ⌉ +1 = n − k − ⌈ k/r ⌉ + 1 . Then By (4), d ⊥⌈ k/r ⌉ = ( n + 1) − g n − k −⌈ k/r ⌉ +1 = k + ⌈ k/r ⌉ . (17)Hence for ⌈ k/r ⌉ ≤ i ≤ n − k , d ⊥ i = k + i . In the meantime, By Lemma 2, It follows that for ≤ i ≤ ⌊ k/r ⌋ , d ⊥ i ≥ d ⊥ i +1 − ( r + 1) . (18)Combing (17) and (18), it follows that for ≤ i ≤ ⌈ k/r ⌉ − , d ⊥ i ≥ d ⊥⌈ k/r ⌉ − ( ⌈ k/r ⌉ − i )( r + 1) = i ( r + 1) − ⌈ k/r ⌉ r + k, hich completes the proof. Remark 4.
Combining Lemma 1, 5 with the condition r | k , we can also obtain the Theorem 2, which appears to be anotherproof of the Theorem 2. Lemma 6.
Let C be an optimal ( n, k, r ) LRC meeting the Singleton-like bound (1) and C ⊥ be its dual code. Then the gapnumbers of the C ⊥ satisfy g ⊥ i ≤ & i + ( ⌈ kr ⌉ r − k ) r ' + i − , ≤ i ≤ k. (19) Proof.
The lemma is proved by induction. For i = k , it follows that g ⊥ k = ( n + 1) − d = k + ⌈ k/r ⌉ − . Suppose that for ≤ i ≤ k , g ⊥ i ≤ l i +( ⌈ kr ⌉ r − k ) r m + i − . Then consider the gap number g ⊥ i − ,1) If l i − ⌈ kr ⌉ r − k ) r m = l i +( ⌈ kr ⌉ r − k ) r m , we have g ⊥ i − ≤ g ⊥ i − l i − ⌈ kr ⌉ r − k ) r m + ( i − − .2) If l i − ⌈ kr ⌉ r − k ) r m = l i +( ⌈ kr ⌉ r − k ) r m − , i.e., ( i −
1) + ( ⌈ kr ⌉ r − k ) = ar, i + ( ⌈ kr ⌉ r − k ) = ar + 1 . By Lemma 5, d ⊥⌈ kr ⌉ = k + ⌈ kr ⌉ and for ≤ i ≤ ⌈ kr ⌉ − , d ⊥ i ≥ i ( r + 1) − (cid:6) kr (cid:7) r + k. Hence the set { , · · · , a ( r + 1) − ( (cid:6) kr (cid:7) r − k ) − } contains at most { d ⊥ , · · · , d ⊥ a − } . In other words, the set { , · · · , a ( r + 1) − ( (cid:6) kr (cid:7) r − k ) − } contains at least thesegap numbers { g ⊥ · · · g ⊥ ar − ( ⌈ kr ⌉ r − k ) } . Then it follows that g ⊥ ar − ( ⌈ kr ⌉ r − k ) ≤ a ( r + 1) − ( (cid:6) kr (cid:7) r − k ) − , i.e., g ⊥ i − ≤ l i − ⌈ kr ⌉ r − k ) r m + ( i − − .Combining the above two cases, the conclusion follows. Theorem 4.
Let C be an optimal ( n, k, r ) LRC meeting the Singleton-like bound (1), its i -th ( ≤ i ≤ k ) GHW d i ≥ n − k − & ⌈ kr ⌉ r − i + 1 r ' + i + 1 . (20) Proof.
For ≤ i ≤ k , the i -th GHW satisfies d i ( a ) = n + 1 − g ⊥ k − i +1( b ) ≥ n + 1 − & k − i + 1 + ⌈ kr ⌉ r − kr ' + k − i ! = n − k − & ⌈ kr ⌉ r − i + 1 r ' + i + 1 , where ( a ) follows from (4) and ( b ) follows from Lemma 6. Remark 5.
Combining Theorem 1, 4 with the condition r | k , Theorem 3 can also be obtained. IV. B
OUNDS OF L INEAR C ODES IN T ERMS OF THE
GHW S In this section, we employ a parity-check matrix approach [3] to present two general bounds of linear codes in terms of theGHWs. The general bounds take the field size into account and can be used to derive several known bounds of LRCs. It isalso shown that the result on the bounds of LRCs in [6] can be explained in terms of the GHWs. Some result in [21] can befurther refined by using the GHWs.
Proposition 1.
Let C be an [ n, k, d ] linear code and C ⊥ be its dual code. The minimum distance of C satisfies d ≤ min ≤ i ≤ g ⊥ k − k d ( q )opt ( n − d ⊥ i , k + i − d ⊥ i ) , (21) where d ( q ) opt ( n ∗ , k ∗ ) is the largest possible minimum distance of a q -ary linear code with length n ∗ and dimension k ∗ , and d ⊥ i is the i -h GHW of C ⊥ .Proof. By (5), we obtain that g ⊥ k − k = max { i : 1 ≤ i ≤ n − k, d ⊥ i < k + i } . (22)n other words, for any ≤ i ≤ g ⊥ k − k , it follows that k + i − d ⊥ i > . Now consider a subcode D ⊥ i of C ⊥ with the support | supp( D ⊥ i ) | = d ⊥ i . Let H = [( h T , · · · h Ti ) T ] be a basis of the subcode D ⊥ i with i independent parity-check equations from D ⊥ i .Now let us select n − k independent parity-check equations from C ⊥ to construct a parity-check matrix H = ( H T , H T ) T of C , where the first i parity-check equations are the vectors from H . The lower part H consists of other n − k − i independent parity-check equations form C ⊥ . The first i rows in H cover | supp( D ⊥ i ) | columns of H . By deleting the first i rows and the corresponding | supp( D ⊥ i ) | columns of H , we have an m ∗ × n ∗ sub-matrix H ∗ , where m ∗ = n − k − i and n ∗ = n − | supp( D ⊥ i ) | = n − d ⊥ i . Let C ∗ be the [ n ∗ , k ∗ , d ∗ ] linear code with parity-check matrix H ∗ . Among the n ∗ columns of H , since the elements lies above H ∗ are all zero, d ≤ d ∗ . Moreover, by rank( H ∗ ) ≤ n − k − i , we have k ∗ = n ∗ − rank( H ∗ ) ≥ k + i − d ⊥ i > . Hence, d ≤ d ∗ ≤ d ( q )opt ( n ∗ , k ∗ ) ≤ d ( q )opt ( n − d ⊥ i , k + i − d ⊥ i ) . (23)Since ≤ i ≤ g ⊥ k − k , the conclusion follows.In the proof of Proposition , the next result follows by substituting (23) by k ≤ k ∗ − i + d ⊥ i ≤ k ( q )opt ( n ∗ , d ∗ ) − i + d ⊥ i ≤ k ( q )opt ( n − d ⊥ i , d ) − i + d ⊥ i . (24) Proposition 2.
Let C be an [ n, k, d ] linear code and C ⊥ be its dual code. The dimension of C satisfies k ≤ min ≤ j ≤ g ⊥ k − k h k ( q )opt ( n − d ⊥ i , d ) − i + d ⊥ i i , where k ( q ) opt ( n ∗ , d ∗ ) is the largest possible dimension of a q -ary linear code with length n ∗ and minimum distance d ∗ and d ⊥ i is the i -h GHW of C ⊥ . Remark 6.
As for ( n, k, r ) LRCs, By Lemma 4, g ⊥ k ≥ ⌈ k/r ⌉ + k − . Hence g ⊥ k − k ≥ ⌈ k/r ⌉ − . By Lemma 1, for ≤ i ≤ ⌊ k/r ⌋ , d ⊥ i ≤ i ( r + 1) . In the above proof, for ≤ i ≤ ⌈ k/r ⌉ − , if we delete i ( r + 1) − | supp( D ⊥ i ) | columns more,then n ∗ = n − i ( r + 1) . In this case Proposition 1 gives the following bound [3] d ≤ min ≤ i ≤⌈ kr ⌉− d ( q )opt ( n − i ( r + 1) , k − ir ) . (25) where d ( q ) opt ( n ∗ , k ∗ ) is the largest possible minimum distance of a q -ary linear code with length n ∗ and dimension k ∗ . Similarly,Proposition 2 gives the Cadambe-Mazumdar bound [4] k ≤ min ≤ i ≤⌈ kr ⌉− h ir + k ( q )opt ( n − i ( r + 1) , d ) i , (26) where k ( q ) opt ( n ∗ , d ∗ ) is the largest possible dimension of an n ∗ -length code given the alphabet size q and distance d ∗ , For a linear code C , by (5)(6), the k -th gap number of C ⊥ g ⊥ k = k + max { i : 1 ≤ i ≤ n − k, d ⊥ i < k + i } , (27) = k + min { i : 1 ≤ i ≤ n − k, d ⊥ i = k + i } − . (28)Hence d = n + 1 − g ⊥ k = n − k − max { i : 1 ≤ i ≤ n − k, d ⊥ i < k + i } + 1 , (29) = n − k − min { i : 1 ≤ i ≤ n − k, d ⊥ i = k + i } + 2 . (30)In the following, we will show that the results on the minimum distances of linear codes in [6][21] can be explained or refinedby using the properties (29)(30) of the gap number g ⊥ k .Tamo et al. introduced a matroid approach to study the optimality of the minimum distance for a linear code in [6]. Let G be the k × n generator matrix of a linear code C . Define the matroid M ([ n ] , rank( . )) , the rank of a set A ⊆ [ n ] is rank( A ) = rank( G A ) , where G A is the submatrix of G indexed by A and rank is the normal operator on linear vectors. Aset B ⊆ [ n ] is said to be a circuit if rank( B ) = |B| − and all its proper subsets are independent. Tamo et al. introduced thenotion of nontrivial union of circuits and proposed the following result on the minimum distance of linear codes. roposition 3. [6, Theorem 2] Let G , M and µ be defined as above, then C has the minimum distance d = n − k − µ + 2 , (31) where µ is the minimum positive integer such that the size of every nontrivial union of µ circuits in M is at least k + µ . According to the definition of the circuits, the columns of G indexed by the coordinates in a circuit must be linearlydependent. Hence it is not hard to see that a circuit corresponds to a parity-check equation in C ⊥ . A nontrivial union [6]of ν circuits corresponds to ν linearly independent parity-check equations, which corresponds to a subcode D ⊥ in C ⊥ withdimension ν . By the definition, the size of every nontrivial union of ν circuits in M is at least k + ν , hence it follows that d ⊥ ν ≥ k + ν. On the other hand, by the generalized Singleton bound, d ⊥ ν ≤ k + ν. Thus d ⊥ ν = k + ν. Then combining thedefinition that µ is the minimal value of ν , we have µ = min { ν : 1 ≤ ν ≤ n − k, d ⊥ ν = k + ν } . (32)Hence by (30), Proposition 3 follows.Wang et al. proposed a framework of regenerating sets [20], [21] which extends the matroid approach in [6] to include thevector case and nonlinear case. For the linear case of this framework, Wang et al. obtain the following result. Proposition 4. [21, Theorem 2] For an [ n, k, d ] linear code, d ≤ n − k − ρ + 1 , (33) where ρ = max { x : Φ( x ) − x < k } , Φ( x ) = min {| ∪ xj =1 R j | : R , · · · , R x has a nontirvial union } . Similarly, according to the definition of the regenerating set and the nontrivial union [21], we can see that a regeneratingset corresponds to a parity-check equation in C ⊥ and a nontrivial union of x regenerating sets corresponds to a subcode D ⊥ in C ⊥ with dimension x . Hence in terms of GHWs, Φ( x ) = d ⊥ x . Then by (5), ρ = max { x : d ⊥ x − x < k, ≤ x ≤ n − k } = g ⊥ k − k. (34)Thus by (29) we can obtain that d = n − k − ρ + 1 , (35)which gives a refined explanation of the Proposition 4 in terms of the GHWs where (33) holds with equality.V. C ONCLUSIONS
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