Long Optimal or Small-Defect LRC Codes with Unbounded Minimum Distances
aa r X i v : . [ c s . I T ] O c t Long Optimal or Small-Defect LRC Codeswith Unbounded Minimum Distances
Hao Chen, Jian Weng and Weiqi Luo ∗ October 22, 2019
Abstract
For a linear locally recoverable code (LRC) with length n , dimen-sion k and locality r its minimum distance d satisfies d ≤ n − k +2 −⌈ kr ⌉ .A code attaining this bound is called optimal. Many families of opti-mal locally recoverable codes have been constructed by using differenttechniques in finite fields or algebraic curves. However in previous con-structions of length n >> q optimal LRC codes minimum distances areonly few constants smaller than 9. No optimal LRC code over a generalfinite field F q with the length n ∼ q and the minimum distance d ≥ F q , for any given r ∈ { , , . . . , q − } and given d satisfying3 ≤ d ≤ min { r + 1 , q + 1 − r } , we construct explicitly an optimal LRCcode with length n = q ( r + 1), locality r and minimum distance d .We also gives an asymptotic bound for q -ary r ≤ q − r -localityLRC codes with small defect s = n − k +2 −⌈ kr ⌉− d are also constructed. ∗ Hao Chen, Jian Weng and Weiqi Luo are with the College of Information Science andTechnology/Cyber Security, Jinan University, Guangzhou, Guangdong Province 510632,China, [email protected], [email protected], [email protected]. The research ofHao Chen was supported by NSFC Grants 11531002. The research of Jian Weng wassupported by NSFC Distinguished Young Scholar Grant 61825203. The research of WeiqiLuo was supported by NSFC Grant 61877029. Introduction
Let F q be finite field with q elements where q is a prime power. For a linearcode C over F q with length n , dimension k and minimum distance d , wedefine the locality as follows. Given a ∈ F q , set C ( i, a ) = { x ∈ C : x i = a } ,where i ∈ { , . . . , n } is an arbitrary coordinate position. C A ( i, a ) is therestriction of C ( i, a ) to the coordinate positions in A ⊂ { , . . . , n } . Thelinear code C is a locally recoverable code with locality r , if for each i ∈{ , . . . , n } , there exists a subset A i ∈ { , . . . , n } − { i } of cardinality at most r such that C A i ( i, a ) ∩ C A i ( i, a ′ ) = ∅ for any given a = a ′ . It was proved aSingleton-like bound for LRC codes in [12] and [22] d ≤ n − k + 2 − ⌈ kr ⌉ , where ⌈ x ⌉ is the smallest integer greater than or equal to x . It is clear that r ≤ k , this upper bound is just the Singleton bound d ≤ n − k + 1 forlinear codes when r = k . A linear code attaining this upper bound is calledan optimal LRC code. In coding theory linear codes attaining the Singletonbound d ≤ n − k + 2 are called MDS (maximal distance separable). Then theoptimal LRC code is a generalization of the MDS (Maximal Distance Sep-arable) code. We refer to [12, 22, 28, 31] for the background in distributedstorage.The main conjecture of MDS codes claims that the length of an MDScode over F q is at most q +1, except some trivial exceptional cases. Many op-timal LRC codes with large code length n > q have been constructed. Hencethe main conjecture type upper bound on the lengths of optimal LRC codesdoes not hold directly. However it is still a challenging problem to ask themaximal possible length of an optimal LRC code over any given finite field F q . We refer to [2, 26] for the discussion of the background. Consideringthe recent progress in [2, 15, 21, 30, 18] it is natural to ask that if thereexists optimal LRC q -codes with length n ∼ q and unbounded locality andunbounded minimum distances. In this paper we give an affirmative answer.On the other hand almost and near MDS codes have been studied incoding theory. Almost MDS codes are linear [ n, k, d ] q codes satisfying n − k + 1 − d = 1. An almost MDS codes with its dual also almost MDS iscalled near MDS. We refer to [6, 5, 7, 3, 14, 9, 8] for almost and near MDS2odes. For near MDS codes it was conjectured that their maximal lengthsare around q + 2 √ q (see [8]). In [9] Corollary 8 it was proved that the length n of an almost MDS [ n, k, n − k ] q code satisfies n ≤ d + 2 q if k ≤ d > q .In this paper we call s = n − k + 2 − ⌈ kr ⌉ − d the defect of the locality- r LRCcode. Many long r -locality LRC codes with very large lengthes and smalldefects are also constructed, see Table 2.The more general locally recoverable codes tolerating multiple erasurescan be defined as follows. A linear code C ⊂ F nq has ( r, µ )-locality if eachcoordinate position i ∈ { , . . . , n } is contained in a subset A ⊂ { , . . . , n } with cardinality r + µ − C A of C to A has minimumdistance at least µ . In the case µ = 2, it is just the LRC code with thelocality r . The Singleton-like bound for a linear [ n, k, d ] q code with ( r, µ )-locality is d ≤ n − k + 1 − ( ⌈ kr ⌉ − µ − r, µ )-locality. Tamo-Barg good polynomial con-struction in [26] of r -LRC codes can be generalized to optimal LRC codeswith ( r, µ )-locality. Some other optimal LRC codes with ( r, µ )-locality wereconstructed in [23, 27, 4, 16]. We summarize previous constructions of optimal LRC codes in [23, 29, 13,15, 25, 26, 27, 2, 20, 19, 4, 16] as follows. n > q Binary LRC codes over F with large lengths: In [29] an al-most optimal binary LRC code with n = 15 , k = 10 , r = 6 and d = 4 < −
10 + 2 − ⌈ ⌉ = 5 was constructed. In [13] a family of optimal bi-nary cyclic LRC codes satisfying n = 2 m − m , r + 1 | n , d = 2 was constructed. In [17] some upper bounds on the minimumdistances of LRC and constructions of binary LRC were given. Many inter-esting construction of LRC with small localities over binary or small fieldswere given in [32]. 3. Optimal LRC codes over F and F with large lengths: In[29] optimal LRC codes over F with n = 4 i + 4 , k = 3 i + 1 , r = 3 , d = 4, i ≥ F with n =18 , k = 11 , r = 2 , d = 3 was constructed. An optimal LRC code over F with n = 24 , k = 17 , r = 3 , d = 3 and some other optimal LRC codes over F and F with length 48 and 110 were also constructed in [2]. It was asked in[2] if there exists a family of optimal LRC codes over F q with length n ∼ q , d = 3 and all values of r . In [21] distance 3 and 4 optimal LRC codes witharbitrary lengths were constructed via cyclic codes. The locality r has tosatisfy some number-theoretic property in the result of [21].3. Optimal LRC codes over F q with lengths up to q + 2 √ q : In[19] by the using of elliptic curves and other algebraic-geometric techniques,optimal LRC codes over F p a with code length up to p a + 2 √ p a and locality r ≤ q = 2 a , a even, the locality of optimalLRC codes in [19] can be 23. To our knowledge this is the only known familyof optimal LRC codes with larger distances over a general finite field withcode lengths greater than field size. However the locality has to be smallerthan or equal to 23.4. More general codes:
In July, 2018 Guruswami, Xing and Chenproved in [15] an upper bound n ≤ O ( dq d − ) on the length n of an opti-mal LRC [ n, k, d ] q code over F q satisfying n ≥ Ω( dr ). They also proved theexistence of optimal LRC codes satisfying n ≥ Ω d,r ( q ⌊ d − ⌋ ). In [30, 18]some optimal LRC with n very close to q and small distances d = 5 , , , n < q In [23] optimal LRC codes with n = ⌈ kr ⌉ ( r + 1) and q > n was constructed.Optimal cyclic LRC codes over any given finite field F q with ( r + 1) | q − n | q − F q with n slightly smaller than q were constructed by theusing of good polynomials. This was extended in [20] to give more suchoptimal LRC codes over any given finite field F q with more possible valuesof the locality. In [27, 4, 16, 10] optimal ( r, µ )-LRC codes with some specialproperties were constructed from cyclic codes. However few known optimal( r, µ )-LRC codes over F q have their lengths larger than q . In all previousconstructions minimum distances of long optimal LRC codes are bounded by ome absolute constant. Main open problem.
In all above cases no optimal LRC code over F q with length n ∼ q and and unbounded minimum distance d ≥ . In [1] Proposition 6.3 the following asymptotic bound for limits R =lim k i n i and δ = lim d i n i for sequences of LRC codes with locality r and lengths n i −→ ∞ was established by using Garcia-Stichtenoth curves. R ≥ rr + 1 (1 − δ − √ q + 3 ) , r = √ q − , and R ≥ rr + 1 (1 − δ − √ qq − , r = √ q. We will give new constructions of LRC codes with any given locality r ≤ q with better asymptotic bound in Section 4. In this paper we prove the following main result.
Main result.
Over any given finite field F q , for a given positive integer r ∈ { , , . . . , q − } , a given positive integer w satisfying w ≤ min { r − , q − − r } , and a positive integer l ≤ q , an optimal LRC [( r + 1) l, rl − w, w + 2] q code with the locality r can be constructed. We also extend our result to optimal ( r, µ )-LRC codes.
Corollary 1.1.
1) For any prime power q ≥ , we construct explicit [ q − q, q − q − , q optimal LRC code with locality r = q − over F q ;2) For any prime power q ≥ , we construct explicit [ q − q, q − q − , q optimal LRC code with locality r = q − over F q ; ) For any prime power q ≥ , we construct explicit [ q − q, q − q − , q optimal LRC code with locality r = q − over F q . Compared with known optimal LRC codes in subsection 1.2.1 our con-struction gives a lot of longer optimal LRC codes with unbounded minimumdistances. This shows that the lengths of optimal LRC codes can be veryclose to q even with unbounded minimum distances. This is quite differentto the main conjecture type upper bound on MDS codes. Open Problem.
From our construction and the result in [15] it is nat-ural to ask if there exist LRC codes with length n ≥ q and unboundedlocalities and unbounded distances.We give a new asymptotic bound as follows. It is better than Proposition6.3 in [1] in some interval of γ . Better asymptotic bound.
Let q be a square of a prime power. Forany given locality r ≤ q − , and any given real number γ ∈ (0 , , we havea sequence of q -ary LRC codes with locality r satisfying δ ≥ (1 − γ )(1 − r − q ) , and R ≥ ( γ − √ q ) rq . Let X be a set and F q be any given finite field. The function g is a function X −→ F q such that there exist l sets A ⊂ g − ( y ) , . . . , A l ⊂ g − ( y l ) ofcardinality | A i | = r + 1, i = 1 , . . . , l , where y , . . . y l ∈ F q are l distinctelements in F q (then l ≤ q ). It is easy to construct the set X and thefunction g satisfying the above property. For example, X = A × F q , where A is a set of cardinality r + 1 and g is the projection to the second factor.For any given a = ( a hs ) ≤ h ≤ r − , ≤ s ≤ t − ∈ F kq , k = rt, t ≤ l , we consider the function F a ( x, y ) = Σ r − h =0 Σ t − s =0 a hs g ( x ) s y h on X × F q . Let B ⊂ F q be a subset with r +1 distinct elements b , b , . . . , b r +1 (then r ≤ q −
1) in F q . We denote r + 1 elements of A i as x i , x i , . . . , x ir +1 , i = 1 , . . . , l , then g ( x ij ) = y i for j = 1 , . . . , r +1. The subset A ⊂ X × F q con-sists of the following ( r + 1) l elements ( x ij , b j ) for 1 ≤ i ≤ l and 1 ≤ j ≤ r + 1. Proposition 2.1.
We assume t ≤ l . If F a ( x, y ) is zero on all points ofthe set A , then a = 0 . Proof.
We consider F a ( x, y ) on the subset consisting of r + 1 elements( x i , b ) , . . . , ( x ir +1 , b r +1 ). Then F a ( x ij , b j ) = Σ r − h =0 (Σ t − s =0 a hs g ( x ij ) s ) b hj . Since g ( x ij ) = y i , this is a constant for all x i , . . . , x ir +1 , set Σ t − s =0 a hs g ( x ij ) s = c h . Then the polynomial Σ r − h =0 c h x h has r + 1 roots b , . . . , b r +1 . This impliesthat Σ t − s =0 a hs y si = 0 for all possible y , . . . , y l . Since t − < l , then theconclusion a hs = 0 follows directly. Set U ⊂ F rtq be a linear subspace with dimension u , we consider the linearcode C ( U ) ⊂ F nq , n = ( r + 1) l , defined by C ( U ) = { ( F a ( x ij , b j )) ≤ j ≤ r +1 , ≤ i ≤ l : a ∈ U } . Since F a + F b = F a + b and F λ a = λF a for λ ∈ F q and a , b in F kq , this is alinear code with dimension u from Proposition 2.1. Definition 2.1.
For any given a ∈ F kq , k = rt , H a is number of commonroots in the set B of t equations Σ r − h =0 a hs b h = 0 for s = 0 , , . . . , t − . Thatis, there exist H a elements of the set B , b i , . . . , b i H a , such that Σ r − h =0 a hs b hi f = 0 for f = 1 , . . . , H a and s = 0 , , . . . , t − . We define H ( U ) = max { H a : a ∈ U } . heorem 2.1. The locality of C ( U ) is at most r , the minimum distanceof C ( U ) is at least n − ( r + 1)( t − − H ( U )( l − t + 1) . Proof.
For a given coordinate position, say ( x i , b ), if F a ( x i , b ) = F a ( x i , b ), then the evaluation vector of F a and F a at coordinate positions( x i , b ) , . . . , ( x ir +1 , b r +1 ) can not be the same. Otherwise F a w ( x ij , b ) = Σ r − h =0 (Σ t − s =0 a whs g ( x ij ) s ) b h = Σ r − h =0 (Σ t − s =0 a whs y si ) b h are the same for w = 1 ,
2. Here we notice that g ( x ij ) = y i for all j =1 , , . . . , r +1. If the evaluation vectors above are the same, since (Σ t − s =0 a whs y si )are constants only depending on i and a w , the two polynomials in b of degree r − r points b , . . . , b r +1 . Then the two polynomials haveto be the same, that is, Σ t − s =0 a hs y si = Σ t − s =0 a hs y si for all h = 0 , . . . , r − F a ( x i , b ) = F a ( x i , b ). On the other hand if the evaluationat the r points ( x i , b ) , . . . , ( x ir +1 , b r +1 ), F a ( x ij , b j ) = Σ r − h =0 (Σ t − s =0 a hs g ( x ij ) s ) b hj are given, then the r coefficients Σ t − s =0 a hs g ( x ij ) s = Σ t − s =0 y si can be solved fromthe Vandermonde matrix. Then the value F a ( x i , b ) = Σ r − h =0 (Σ t − s =0 a hs y si ) b h can be recovered. Here we notice that g ( x i ) = y i from the definition of thefunction g . This is essentially the same as the recover procedure in page4663 of [26]. Thus the locality is at most r .For any given a = ( a hs ) ≤ h ≤ r − , ≤ s ≤ t − ∈ U ⊂ F rtq , we consider F ( x ij , b j ) = Σ t − s =0 (Σ r − h =0 a hs ( b j ) h ) g ( x ij ) s . From Definition 2.1 the equation Σ r − h =0 a hs ( b j ) h = 0 for s = 0 , . . . , t −
1, isvalid for H a ≤ r − b i , . . . b i H a in the set B = { b , . . . , b r +1 } . Thenthe number of zeros of F in the set A is at most H a l + ( r + 1 − H a )( t −
1) = H a ( l − t + 1) + ( r + 1)( t − . Actually for each b i , , . . . , b i H a , there are l solutions. For any element b j inthe set B − { b i , . . . , b i H a ), F ( x ij , b j ) = Σ t − s =0 (Σ r − h =0 a hs ( b j ) h ) g ( x ij ) s = Σ t − s =0 (Σ r − h =0 a hs ( b j ) h ) y si is not a zero polynomial, then it has at most t − y i satisfying Σ t − s =0 (Σ r − h =0 a hs ( b j ) h ) y si = 0 . y i , g ( x ij ) = y i has at most one solution x ij , since j is fixed.The conclusion is proved. d = 2This case is trivial. In the case U = F rtq , it is clear H ( U ) = r − d = n − ( r + 1) t + 2 − (( r − l − t + 1) − r + 1). When t < l , d = n − k + 2 − ⌈ rt/r ⌉ − (( r − l − t + 1) − r + 1) < n − k + 2 − ⌈ rt/r ⌉ .When t = l , d = n − ( r + 1) t + 2. That is we have a length n = ( r + 1) l ,dimension k = rl , minimum distance d = 2 optimal LRC code over F q withany given locality r ≤ q − Let F q be a finite field satisfying q > r + 1 r − w ! . For r + 1 r − w ! linear subspaces W i × · · · × × W i ( l copies) in P r − ( F q ) × · · · × P r − ( F q ) ( l copies), where W i is a dimension w − linear subspace in P r − ( F q ) , i = 1 , . . . , r + 1 r − w ! , we can find a codimension w linear subspace U in P r − ( F q ) × · · · × P r − ( F q ) ( l copies) such that the intersection of U withthe union of these dimension v linear subspaces is empty. Proof.
Suppose linear independent vectors have been chosen, in thefinal step, if r + 1 r − w ! q r − < q r , then we can find the desired linear indepen-dent vector in F rq . The conclusion is proved.It should be noticed that when q is small, the conclusion of Lemma 3.1is not valid. For example if r − > q , P r − ( F q ) is covered by r ( r +1)2 hyper-planes defined x i = x j , where i = j .In the above construction if U is the full space F rtq , it is clear that H ( U )can attain the maximal possibility r −
1, that is, for some a = ( a hs ), thereare r − b in the set { b , . . . , b r +1 } , such that Σ r − h =0 a hs b h = 0 forall s = 0 , , . . . , t −
1. Then in this case the minimum distance of the con-9tructed optimal LRC code is d = 2. Then we show that d of the constructedoptimal LRC code can be enhanced if q > r + 1 ,r + 2 − d ! is satisfied.For a equation Σ r − h =0 c h b h = 0 , where ( c , c , . . . , c r − ) are r − F q considered asa point in the projective space P r − ( F q ), if r − c , c , . . . , c r − ) is a fixed point in P r − ( F q ). Then for r ( r +1)2 possibilitiesof r − B = { b , . . . , b r +1 } , they corresponds to r ( r +1)2 points of coefficients in P r − ( F q ) × · · · × P r − ( F q ) ( l copies) satisfying thatΣ r − h =1 c hs b hi j = 0 for s = 0 , , . . . , l − j = 1 , . . . , r −
1, where ( b i , . . . , b i r − )is any fixed r − B . From Lemma 3.1 if q > r + 1 r − ! ,there exists a codimension 1 linear subspace U of P r − ( F q ) × · · · × P r − ( F q )( l copies) such that these r ( r +1)2 coefficient points are not in U . That is for a ∈ U ⊂ F rlq , at least for one of s ∈ { , , . . . , t − } ,Σ r − h =0 a hs b h = 0can not have r − B . Hence H ( U ) in Theorem 2.1 cannot be its maximal possibility r −
1, we have H ( U ) ≤ r −
2. From U wehave a linear code with length n = ( r + 1) l , dimension k = rl − d ≥ n − ( r − l − ( r + 1 − r + 2)( l −
1) = 3. This code has locality r and satisfies the Singleton-like bound. It is an optimal LRC code with n = ( r + 1) l, k = rl − , d = 3 and locality r . In general we consider the case that the equationΣ r − h =0 c h b h = 0 , where ( c , c , . . . , c r − ) are r − F q considered as apoint in the projective space P r − ( F q ), has at least r − w roots in the set B ,where w is a fixed positive integer in the set { , , . . . , r − } . For example,suppose b , . . . , b r − w are such r − w roots. Then the equation is of the form C ( x − b ) · · · ( x − b r − w )( x w − + c ′ w − x w − + · · · + c ′ x + c ′ )10 where C, c ′ w − , . . . , c ′ , c ′ are w + 1 variables. Then the coefficient points( c , . . . , c r − ) corresponds to a linear subspace in P r − ( F q ) of dimension w −
1. We have r + 1 r − w ! such linear subspaces in P r − ( F q ) × · · · × P r − ( F q )( l copies). Hence from Lemma 3.1 if q > r + 1 r − w ! , a linear subspace in P r − ( F q ) ×· · ·× P r − ( F q ) ( l copies) with codimension w can be found whichhas no intersection with these r + 1 r − w ! products of such linear subspaces.That is, we can find a linear subspace U w of P r − ( F q ) × · · · × P r − ( F q ) ( l copies) of codimension w such that for a ∈ U w ⊂ F rlq , at least for one of s ∈ { , , . . . , t − } , Σ r − h =0 a hs b h = 0can not have r − w roots in the set B . Then H ( U w ) ≤ r − w −
1. TheSingleton-like upper bound is d ≤ ( r +1) l − ( rl − w )+2 −⌈ rl − wr ⌉ = w +2. FromTheorem 2.1, the lower bound is d ≥ ( r +1) l − ( r +1)( l − − ( r − w −
1) = w +2.Hence the locality is r and d = w + 2.Therefore we have proved that the minimum distance of the constructedoptimal LRC code can be enhanced when the field size is large. In the fol-lowing part we prove that actually the same conclusion can be proved whenthe field size satisfies q ≥ r + d − The above construction depends on Lemma 3.1 with a ”counting points”argument. However the linear subspace of P r − ( F q ) (of the coefficients( c , . . . , c r − ) ) defined by the condition that ”there are r − w roots b i , . . . , b i r − w in the set B ” is defined by a ( r − w ) × r partial Vandermonde matrix asfollows. b i · · · b r − i b i · · · b r − i · · · · · · · · · · · · b i r − w · · · b r − i r − w Hence if r + 1 + w ≤ q is satisfied, we can pick up w distinct elements e , . . . , e w in F q − B . Then the codimension w linear subspace U in P r − ( F q )11efined by the following partial w × r Vandermonde matrix satisfying therequirement in Lemma 3.1. Actually a r × r Vandermonde matrix from r distinct elements is of rank r . e · · · e r − e · · · e r − · · · · · · · · · · · · e w · · · e r − w The two conditions about the coefficient vector that1)there are r − w roots in the set B and2)in the subspace U ,correspond to a rank r Vandermonde r × r matrix. If the linear subspace U has non-empty intersection with one of the r + 1 r − w ! linear subspaces inLemma 3.1, the point in the intersection has to be a zero vector. This is acontradiction. Then we have the following result. Theorem 2.2.
For any given finite field F q , a positive integer r ∈{ , , . . . , q − } , a positive integer ≤ w ≤ q − − r , and a positive integer l ≤ q , an optimal LRC [( r + 1) l, rl − w, w + 2] q code with the locality r canbe constructed. In the following Table 1 we give many long optimal LRC codes with n very close to q . Table 1
Explicit optimal LRC codes with n ∼ q locality length dimension distance q − q − q q − q − q − q − q q − q − q − q − q q − q − q − q − q q − q − q − q − q q − q − q − q − q q − q − q − q − q q − q −
10 12 q − q − q q − q −
11 1312
Long optimal ( r, µ ) -LRC codes Let µ ≥ g as in section 2and pick up l subsets A i ⊂ g − ( y i ), where y , . . . , y l are distinct l elementsin F q , | A i | = r + µ − i = 1 , , . . . , l . For any given a = ( a hs ) ≤ h ≤ r − , ≤ s ≤ t − ∈ F kq , where k = rt, t ≤ l , we consider the function F a ( x, y ) = Σ r − h =0 Σ t − s =0 a hs g ( x ) s y h on X × F q . Let B ⊂ F q be a subset with r + µ − b , b , . . . , b r + µ − , then r + µ ≤ q . We denote r + µ − A i as x i , x i , . . . , x ir + µ − , i = 1 , . . . , l . The subset A ⊂ X × F q consists of thefollowing ( r + µ − l elements ( x ij , b j ) for 1 ≤ i ≤ l and 1 ≤ j ≤ r + µ − U ⊂ F rtq be a linear subspace with dimension u , we consider the linearcode C ( U ) ⊂ F nq , where n = ( r + µ − l , defined by C ( U ) = { ( F a ( x ij , b j ) : i = 1 , . . . , l, j = 1 , . . . , r + µ −
1) : a ∈ U } . This is a linear code with dimension u . We have the following result. Theorem 3.1. C ( U ) is a ( r, µ ) -LRC code, the minimum distance of C ( U ) is at least n − ( r + µ − t + µ − ( H ( U )( l − t + 1) − r + 1) . Proof.
We consider the restriction of C ( U ) to the subset B i = { ( x i , b ) , . . . , ( x ir + µ − , b r + µ − ) } Then the conclusion follows from a similar argument as the proof of Theo-rem.2.1.When t = l, U = F rlq , then H ( U ) = r −
1. Hence we get a ( r, µ )-LRC codeattaining the Singleton-like bound, with length n = ( r + µ − l, k = rl, d = µ .Hence for any given finite field F q , a positive integer l ≤ q , a locality ( r, µ )where r is any value in { , , . . . , q − } and µ is an arbitrary positive inte-ger satisfying 2 ≤ µ ≤ q + 1 − r , an optimal ( r, µ )-LRC code with length n = ( r + µ − l , dimension k = rl and minimum distance d = µ can beconstructed. 13or optimal ( r, µ )-LRC codes we have the following result by a similarconstruction as in the proof of Theorem 2.2. Corollary 3.1.
For any given finite field F q , any given ( r, µ ) satisfying r + µ ≤ q + 1 , a positive integer w satisfying ≤ w ≤ q + 1 − r − µ , a positiveinteger l ≤ q , an optimal ( r, µ ) -LRC code over F q with length ( r + µ − l ,dimension rl − w , and distance w + µ can be explicitly constructed. Hence many optimal ( r, µ )-LRC codes with lengths n ∼ q are con-structed. Let X be a smooth projective absolutely irreducible curve of genus g definedover F q , P = { P , . . . , P n } be a set of n F q -rational points, G be a F q -rational divisor with its degree satisfying deg( G ) < n . Let f , . . . , f t − bethe base of L ( G ) = { f : div ( f ) + G ≥ } , where t = deg( G ) − g + 1. Forany given value r ≤ q , we consider functions F ( a , x, b ) = Σ r − h =0 Σ t − s =0 a hs f s ( x ) b h , defined on X × F q where ( x, b ) ∈ X × F q and a = ( a hs ) ≤ h ≤ r − , ≤ s ≤ t − ∈ F rtq .For a given linear subspace U of dimension u in F rtq , we define a linear code C ( U ) = { ( F ( a , x, b )) : ( x, b ) ∈ P × F q , a ∈ U } . Since r − ≤ q anddeg( G ) < n , the evaluation mapping on P × F q is injective, this is a dimen-sion u linear code with length nq and dimension k = (deg( G ) − g + 1) r . Theorem 4.1.
The minimum distance of C ( U ) is at least ( n − deg( G ))( q − r + 1) . Suppose r ≤ q − , this code C ( U ) has locality r . Proof . For each F ( a , x, b ) = Σ r − h =0 Σ t − s =0 a hs f s ( x ) b h = Σ r − h =0 (Σ t − s =0 a hs f s ( x )) b h for x ∈ P and b ∈ F q , there are at most deg( G ) points in P such that thecoefficients Σ t − s =0 a hs f s ( x ) are all zero. For the remaining n − deg( G ) points in P , the degree r − F ( a , x, b ) of b has at most r − F q . Then F ( a , x, b ) as a whole has at most deg( G ) q + ( n − deg( G ))( r − P × F q . The minimum distance is at least nq − deg( G ) q − ( n − deg( G ))( r −
1) = ( n − deg( G ))( q − r + 1) . P , b ) ∈ P × F q , we pick up r distinct points( P , b ) , . . . , ( P , b r ) in P × | bf F q , where b, b , . . . , b r are r + 1 distinct el-ements of F q since r ≤ q −
1. If two functions F ( a , x, b ) and F ( a , x, b )are the same at the r points ( P , b ) , . . . , ( P , b r ), then two degree r − F ( a , P , b ) = Σ r − h =0 (Σ t − s =0 a hs f s ( P )) b h and F ( a , P , b ) = Σ r − h =0 (Σ t − s =0 a hs f s ( P )) b h of b are the same at r elements b , . . . , b r of F q . Hence they have the samecoefficients Σ t − s =0 a hs f s ( P ) = Σ t − s =0 a hs f s ( P ) . Therefore F ( a , P , b ) = F ( a , P , b ) . The recover procedure is the same as the proof of Theorem 2.1. The con-clusion is proved.We take the family of Garcia-Stichtenoth curves X m of genus g m −→ ∞ over F q where q is a square of a prime power,lim g m N m = √ q. Here N m is the number of F q rational points of X m . In Theorem 4.1 set P m as the set of all these F q rational points. Then we have a family of r -localityLRC codes for any given r ≤ q − n m = N m q , dimension k m =(deg( G m ) − g m +1) r and minimum distance d m ≥ ( N m − deg( G m ))( q − r +1).We take a sequence of F q rational divisors G m such that lim deg( G m ) N m = γ .The conclusion of the following Corollary 4.1 follows directly. Corollary 4.1
Let q be a square of a prime power. For any given fixedlocality r ≤ q − , and any given real number γ ∈ (0 , , we have a sequenceof q -ary LRC codes with locality r satisfying δ ≥ (1 − γ )(1 − r − q ) , and R ≥ ( γ − √ q ) rq . Long LRC codes with small defects
In the construction if we replace F q by a subset S ⊂ F q of r + 1 elementsthe following result follows directly. Theorem 5.1. If r ≤ q − is fixed, we construct an explicit length n ( r + 1) , dimension (deg( G ) − g + 1) r , minimum distance d ≥ n − deg( G )) r -locality LRC code. The defect is s = ( n − deg( G ))( r − g − r +1)+2 . Corollary 5.1.
From a elliptic curve with N rational points over F q and a fixed r ≤ q − and an integer ≤ t ≤ N , we construct r -locality LRCcodes with length ( r + 1) N , dimension tr , minimum distance N − t ) . Thedefect s = N − t + 2 . It follows Theorem 5.1 we have the following asymptotic bound for r -locality LRC codes with reasonable small defect limit S = lim sn ( r +1) . Corollary 5.2
Let q be a square of a prime power. For any given fixedlocality r ≤ q , and any given real number γ ∈ (0 , , we have a sequence of q -ary LRC codes with locality r satisfying δ ≥ − γ ) r + 1 ,R ≥ ( γ − √ q ) rr + 1 and S ≤ (1 − γ ) r − r + 1 + 1 √ q . In the following Table 2 r -locality LRC codes over small fields with largelengths and small defects are given. Algebraic curves with many rationalpoints are from [11]. Table 2
Explicit r -locality LRC codes over F q
16 locality length dimension distance defect3 2 21 8 6 53 2 24 8 6 83 2 30 8 8 123 2 36 10 8 154 2 27 8 10 74 2 30 8 10 168 3 56 36 4 68 3 96 60 4 148 4 70 48 4 68 4 120 80 4 169 3 64 42 4 69 3 64 36 8 109 3 112 69 6 1616 3 100 60 10 727 3 152 90 16 1032 3 176 120 8 664 3 324 232 8 681 3 400 270 20 12128 3 600 420 20 12
In this paper for any given finite field F q , any given r ∈ { , , . . . , q − } andgiven d satisfying 3 ≤ d ≤ min { r + 1 , q + 1 − r } , we give an optimal LRCcode with length n = q ( r + 1), locality r and minimum distance d . Thisis the only known family of optimal LRC codes with n ∼ q , unboundedlocalities and unbounded distances. We speculate there exist optimal LRCcodes with n ≥ q , unbounded localities and unbounded distances. By theusing of Garcia-Stichtenoth curves a better asymptotic bound for r -localityLRC codes is presented. We also construct many long r -locality LRC codeswith small defects. References [1] A. Barg, I. Tamo and S. Vlˇadut¸, Locally recoverable codes on algebraiccurves, IEEE Transactions on Information Theory, vol.63, no. 8, pp.4928-4939, 2017. 172] A. Barg, K. Haymaker, E. W. Howe, G. Matthews and A. V´arilly-Alvarado, Locally revoerable codes from algebraic curves and surfaces,in Algebraic Geometry and Coding Theory and Cryptography, E. W.Howe, K. E. Lauter and J. L. Walker, Editors, Springer, pp. 95-126,2017.[3] M. A. De Boer, Almost MDS codes, Designs, Codes and Cryptography,vol. 9, pp. 143-155, 1996.[4] B. Chen, S. T. Xia, J. Hao and F. W. Fu, Construction of optimalcyclic ( r, δ ) locally repairable codes, IEEE Transactions on InformationTheoru, vol. 64, no. 4, pages 2499-2511, 2018.[5] S. Dodunekova, S. M. Dodunekove and T. Klove, Almost-MDS andnear-MDS codes for error detection, IEEE Transactions on InformationTheory, vol. 43, no. 1, pp. 285-290, 1997.[6] S. Dodunekov and I. Landgev, On near-MDS codes, Journal of Geom-etry, vol. 54, no. 1-2, pp. 30-42, 1995.[7] S. Dodunekov and I. Landgev, Near-MDS codes over some small fields,Discrte Math., vol. 213, pp. 55-65, 2000.[8] S. M. Dodunekov, Applications of near MDS codes in cryptography,ARW, V. Tarnova, 6-9 October, 2008.[9] A. Faldum and W. Willems, Codes with small defects, Designs, Codesand Cryptography, vol. 10, pp. 341-350, 1997.[10] W. Fang and F. Fu, Optimal cyclic ( r, δ ∼∼