Higher Hamming weights for locally recoverable codes on algebraic curves

Contents lists available atScienceDirect

Finite

Fields

and

Their

Applications

www.elsevier.com/locate/ffa

Higher

Hamming

weights

for

locally

recoverable

codes

on

algebraic

curves

Edoardo Ballico1_{, Chiara Marcolla}∗

Department of Mathematics, University of Trento, Italy

a r t i c l e i n f o a b s t r a c t

Article history:

Received18May2015 Receivedinrevisedform30 November2015

Accepted11March2016 Availableonline25March2016 CommunicatedbyChaopingXing

MSC:

primary11G20

secondary11T71,14H51,14H50

Keywords:

AlgebraicgeometricLRCcodes HigherHammingweights Norm-TraceLRCcodes

We study locally recoverable codes on algebraiccurves. In thefirstpart ofthemanuscript,weprovideaboundon the generalizedHamming weight of these codes. In the second part,we propose a new familyof algebraic geometricLRC codes, which are LRC codes from the Norm-Trace curve. Finally,usingsomepropertiesofHermitiancodes,weimprove theboundsonthedistanceproposedinBargetal.(2015)[1]

ofsomeHermitianLRCcodes.

© 2016ElsevierInc.All rights reserved.

1. Introduction

Thev-thgeneralizedHammingweightdv(C) of alinearcodeC istheminimum

sup-portsizeof v-dimensionalsubcodes ofC.Thesequenced1(C),. . . ,dk(C) ofgeneralized

HammingweightswasintroducedbyWei[37]tocharacterizetheperformanceofalinear codeonthewire-tapchanneloftypeII.Later,theGHWsoflinearcodeshavebeenused

* Correspondingauthor.

E-mail addresses:ballico@science.unitn.it(E. Ballico),chiara.marcolla@unito.it(C. Marcolla).

1

ThefirstauthorispartiallysupportedbyMIURandGNSAGAofINdAM (Italy).

http://dx.doi.org/10.1016/j.ffa.2016.03.004

inmanyotherapplicationsregardingthecommunications,asforboundingthecovering radius of linearcodes [15],innetwork coding [26], inthecontext of listdecoding [7,9], andfinallyforsecuresecretsharing[18].Moreover,in[2]theauthorsshowinwhichway an arbitrarylinearcodegivesrise toasecretsharingscheme, in[16,17]theconnection betweenthetrellisorstatecomplexityofacodeanditsGHWsisfoundandin[4]the au-thorprovestheequivalencetothedimension/lengthprofileofacodeanditsgeneralized Hammingweight.Forthesereasons,theGHWs(andtheirextended version, therelative

generalizedHammingweights[21,19])playacentralroleincodingtheory.Inparticular, generalizedandrelativegeneralizedHammingweightsarestudiedforReed–Mullercodes

[10,23]andforcodesconstructedbyusinganalgebraiccurve[6]asGoppacodes[24,38], Hermitian codes[12,25]andCastlecodes[27].

In this paper, we provide a bound on the generalized Hamming weight of locally recoverablecodesonthealgebraic curvesproposedin[1].Moreover,weintroduceanew family of algebraic geometric LRC codes and improve the bounds on the distance for someHermitianLRCcodes.

Locally recoverable codes were introduced in [8] and they have been significantly studiedbecauseof theirapplications indistributedand cloudstoragesystems[3,13,32, 34,35]. Werecall thatacodeC ∈ (Fq)n haslocality r ifeverysymbol of acodewordc

canberecovered fromasubsetofr othersymbolsof c.

Inother words,weconsider afinitefieldK =Fq, whereq isapowerofaprime,and

an[n,k] codeC over thefieldK,where k = log_q(|C|).Foreachi∈ {1,. . . ,n} andeach

a∈ K setC(i,a)={c∈ C | ci= a}.ForeachI⊆ {1,. . . ,n} andeachS⊆ C letSI be

therestrictionofS to thecoordinatesinI.

Definition 1. LetC be an[n,k] code overthe fieldK, where k = log_q(|C|). Then C is

said to have all-symbol locality r if for each a ∈ Fq and each i ∈ {1,. . . ,n} there is

Ii ⊂ {1,. . . ,n}\ {i} with|Ii| r, suchthatforCIi(i,a)∩ CIi(i,a)=∅ forall a= a.

Weusethenotation(n,k,r) torefertotheparametersof thiscode.

Notethatifwereceiveacodewordc correctexceptforanerasureati,wecanrecover thecodewordbylookingatitscoordinatesinIi.Forthisreason,Iiiscalledarecovering

set for thesymbol ci.

Let C be an (n,k,r) code, then the distance of this code has to verify the bound proved in[28,8] thatisd n− k − k/r + 2.Thecodes thatachievethis boundwith equality arecalled optimal LRCcodes [32,34,35].Note thatwhenr = k,we obtainthe Singletonbound, thereforeoptimalLRCcodeswithr = k are MDScodes.

Layout ofthepaper Thispaperisdividedasfollows.InSection2werecallthenotions ofalgebraicgeometriccodesandthedefinitionofalgebraicgeometriclocallyrecoverable codes introducedin [1].In Section3we provide abound on thegeneralized Hamming weights ofthelattercodes.InSection4weproposeanewfamilyofalgebraicgeometric LRC codes,whichare LRCcodes from the Norm-Tracecurve. Finally, inSection5we

improvetheboundsonthedistanceproposedin[1]forsomeHermitianLRCcodes,using somepropertiesoftheHermitiancodes.

2. Preliminarynotions

2.1. Algebraicgeometriccodes

Let K = Fq be a finite field, where q is a power of a prime. Let X be a smooth

projective absolutely irreducible nonsingular curve over K. We denote by K(X ) the

rational functions field on X . Let D be a divisor on the curve X . We recall that the

Riemann–Rochspace associatedtoD isavectorspaceL(D) overK definedas

L(D) = {f ∈ K(X ) | (f) + D  0} ∪ {0},

wherewedenote by(f ) thedivisoroff .

AssumethatP1,. . . ,Pn arerational points onX and D isadivisor such thatD =

P1+ . . . + Pn.LetG besomeother divisorsuchthatsupp(D)∩ supp(G)=∅.Then we

candefine thealgebraicgeometriccodeasfollows:

Definition 2.The algebraic geometric code (or AGcode) C(D,G) associated with the divisorsD andG isdefinedas

C(D, G) ={(f(P1), . . . , f (Pn))| f ∈ L(G)} ⊂ Kn.

ThedualC⊥(D,G) ofC(D,G) isanalgebraicgeometriccode.

In other words an algebraic geometric code is the image of the evaluation map Im(evD)= C(D,G),wheretheevaluationmap evD:L(G)→ Kn isgivenby

evD(f ) = (f (P1), . . . , f (Pn))∈ Kn.

NotethatifD = P1+ . . . + Pn andwedenotebyP = {P1,. . . ,Pn} wecanalsoindicate

evD asevP.

2.2. Algebraicgeometriclocallyrecoverable codes

Inthissectionweconsidertheconstructionofalgebraicgeometriclocallyrecoverable codesof[1].

LetX andY besmoothprojectiveabsolutelyirreduciblecurvesoverK.Letg :X → Y

be arational separable map of curvesof degree r + 1.Since g is separable,then there existsafunctionx∈ K(X ) suchthatK(X )= K(Y)(x) andthatx satisfiestheequation

xr+1+ brxr+ . . . + b0= 0,wherebi∈ K(Y).Thefunctionx canbeconsideredasamap

We consider asubsetS ={P1,. . . ,Ps}⊂ Y(K) of Fq-rational pointsof Y, adivisor

Q∞ suchthatsupp(Q∞)∩ supp(S)=∅ andapositivedivisorD = tQ∞.Wedenote by

A = g−1_{(S) ={P}

ij, where i = 0, . . . , r, j = 1, . . . , s} ⊂ X (K),

where g(Pij) = Pi for all i, j and assume that bi are functions in L(niQ∞) for some

naturalnumbersni withi= 1,. . . ,r.

Let{f1,. . . ,fm} beabasisoftheRiemann–RochspaceL(D).BytheRiemann–Roch

Theorem wehavethatm deg(D)+ 1− g_Y, whereg_Y isthegenusofY.

From now on, we assume that m = deg(D)+ 1− g_Y, where deg(D) = t, and we consider theK-subspaceV ofK(X ) ofdimensionrm generatedby

B = {fjxi, i = 0, . . . , r− 1, j = 1, . . . , m}.

Weconsidertheevaluationmapev_A: V → K(r+1)s_{.Thenwehavethefollowingtheorem.} Theorem 1. The linear space C(D,g) = Span_K(r+1)sevA(B) is an (n,k,r) algebraic

geometric LRCcodewithparameters n = (r + 1)s

k = rm r(t + 1 − g_Y)

d n − t(r + 1) − (r − 1)h.

Proof. SeeTheorem 3.1of[1]. 2

The AGLRCcodes have anadditionalproperty. Theyare LRCcodes (n,k,r) with

(r + 1)|n andr|k.Theset{1,. . . ,n} canbedividedinton/(r + 1) disjoint subsetsUj

for1 j  s withthesamecardinalityr +1.Foreachi thesetIi⊆ {1,. . . ,n}\{i} isthe

complementofi intheelementofthepartitionUjcontainingj,i.e.foralli,j∈ {1,. . . ,n}

eitherIi= Ij orIi∩ Ij=∅.

Moreover, theyhave also the following nice property. Fix w ∈ (K)n _{and denote by}

wUj ={wι,foranyι∈ Uj}.SupposewereceiveallthesymbolsinUj.Thereisasimple

linearparitytestonther + 1 symbolsofUj suchthatifthisparitycheckfailsweknow

that at least oneof thesymbols inUj is wrong. If we are guaranteed (or we assume)

thatatmostoneofthesymbolsinUj iswrongandtheparitycheckisOK,thenallthe

symbols inUj are correct. Moreover we canrecover an erased symbol wι, withι ∈ Uj

using apolynomialinterpolationthroughthepointsoftherecoveringsetwUj.

3. GeneralizedHammingweightsofAGLRCcodes

Let K beafieldand letX beasmoothandgeometrically connectedcurve ofgenus

g  2 defined over the field K. We also assume X (K) = ∅. We recall the following definitions:

Definition 3. (See [29,30].) The K-gonality γK(X ) of X over afield K is the smallest

possibledegreeofadominantrationalmapX → P_K1.ForanyfieldextensionL ofK,we definealsotheL-gonality γL(X ) ofX asthegonalityofthebaseextensionXL=X ×KL.

ItisaninvariantofthefunctionfieldL(X ) ofXL.

Moreover,foreachintegeri> 0,thei-thgonality γi,L(X ) ofX istheminimaldegree

z such thatthere is R∈ Picz(X )(L) with h0_{(R)  i+ 1.The sequence γ}

i,K(X ) is the

usualgonality sequence [20]. Moreover,theintegerγ1,K(X )= γK(X ) istheK-gonality

ofX .

LetK =Fq afinitefieldwithq elements.LetC⊂ Kn bealinear[n,k] codeoverK.

Werecallthatthesupport ofC isdefinedas follows

supp(C) ={i | ci= 0 for some c ∈ C}.

Sosupp(C) is thenumberof nonzerocolumns inagenerator matrixforC. Moreover, for any 1 v  k, the v-th generalized Hamming weight of C [14, §7.10], [36, §1.1] is definedby

dv(C) = min{supp(D) | D is a linear subcode of C with dim(D) = v}.

Inotherwords, foranyinteger1 v  k,dv(C) isthev-thminimumsupportweights,

i.e. the minimal integer t such that there are an [n,v] subcode D of C and a sub-set S⊂ {1, . . . , n} such that (S) = t and each codeword of D has zero coordinates outside S. Thesequence d1(C),. . . ,dk(C) of generalizedHammingweights (alsocalled

weighthierarchy ofC)isstrictlyincreasing(seeTheorem 7.10.1of[14]).Notethatd1(C) istheminimumdistanceofthecodeC.

LetusconsiderX andY smoothprojectiveabsolutely irreduciblecurvesoverK and

letg :X → Y be arational separablemap ofcurvesof degreer + 1. Moreoverwetake

r, t,Q_∞, f1,. . . ,fm and A = g−1(S) defined as Section2.2. So we can construct an

(n,k,r) algebraic geometric LRC code C as in Theorem 1. For this code we have the following:

Theorem 2. Let C be an (n,k,r) algebraic geometric LRC code as in Theorem 1. For every integerv 2 wehave that

dv(C) n − t(r + 1) − (r − 1)h + γv−1,K(X ).

Proof. Takeav-dimensionallinearsubspaceD ofC andcall

E⊆ {Pij | i = 0, . . . r, j = 1, . . . , s},

thesetofcommonzerosofallelementsofD. Sincen− dv(C)= (E),wehavetoprove

zeros of u.Note thatFu is contained inthe set {Pij | i= 0,. . . r,j = 1,. . . ,s} by the

definition of the code C. We have Fu ⊇ E. By the definition of the integers t,  and

h := deg(x),we have (Fu)  t(r + 1)+ (r− 1)h. The divisorsFu− E,u ∈ D \ {0}

form afamilyoflinearlyequivalentnon-negative divisors,eachofthem definedoverK.

Since dim(D) = v,the definition of γ_v−1,K(X ) gives (Fu)− (E)  γv−1,K(X ). This

inequalityforasingleu∈ D \ {0} provesthetheorem. 2 SeeRemark 1foranapplicationof Theorem 2.

4. LRCcodesfromNorm-Tracecurve

InthissectionweproposeanewfamilyofAlgebraicGeometricLRCcodes,thatis,a LRC codes from theNorm-Trace curve. Moreover, wecompute theFqu-gonalityof the

Norm-Trace curve.

Let K =Fqu be afinite field,where q isa powerof aprime.Weconsider the norm

NFqu

Fq andthetrace Tr

Fqu

Fq , twofunctionsfromFqu toFq definedas

NFqu Fq (x) = x 1+q+···+qu−1 and TrFqu Fq (x) = x + x q_{+· · · + x}qu−1_.

TheNorm-Tracecurve χ isthecurvedefinedoverK bythefollowingaffineequation NFqu Fq (x) = Tr Fqu Fq (y), thatis, x(qu−1)/(q−1)= yqu−1+ yqu−2+ . . . + y where x, y∈ K. (1) TheNorm-Tracecurveχ hasexactlyn= q2u−1_{K-rationalaffinepoints(seeAppendixA} of[5]),thatwedenotebyPχ={P1,. . . ,Pn}.Thegenusofχ isg =1₂(qu−1−1)(q

u₋₁ q−1 −1).

Note thatifweconsider u= 2, weobtaintheHermitian curve.

Starting from theNorm-Tracecurve, wehavetwo differentways toconstruct Norm-Trace LRCcodes.

Projection on x Wehave to construct a qu_{-ary (n,k,r) LRC codes.We consider the}

naturalprojectiong(x,y)= x.Thenthedegreeofg isqu−1 = r + 1 andthedegreeofy

is h= 1+ q +· · · + qu−1.

To construct the codes we consider S = Fqu and D = tQ_∞ for some t  1. Then,

using aconstruction of Theorem 1 we find theparameters for these Norm-Trace LRC codes.

Proposition 1.Afamily of Norm-TraceLRCcodeshas thefollowingparameters: n = q2u−1, k = mr = (t + 1)(qu−1− 1)

and

d n − tqu−1− (qu−1− 1)(1 + q + · · · + qu−1).

Projection on y We have to construct a qu_{-ary (n,k,r) LRC codes.We consider the}

othernaturalprojectiong(x,y)= y.Thendeg(g)= 1+ q +· · · + qu−1= r + 1.Inthis casewetakeS =Fqu\M,where

M ={a ∈ Fqu | aq u−1

+ aqu−2+ . . . + a = 0},

so r = q +· · · + qu−1 _{and h = deg(x) = q}u−1_{. Then, using Theorem 1 we have the}

following

Proposition2.A familyof Norm-TraceLRCcodeshas thefollowingparameters: n = q2u−1− qu−1, k = mr = (t + 1)(q +· · · + qu−1)

and

d n − tqu−1− (q + · · · + qu−1)− qu−1(qu−1+· · · + q − 1). FortheNorm-Tracecurveχ weareableto findtheK-gonality ofχ.

Lemma 1. Let χ be a Norm-Trace curve defined over Fqu, where u  2. We have

γ_1,Fqu(χ)= qu−1_.

Proof. The linear projection onto the x axis has degree qu−1 and it is defined over Fq and hence over Fqu. Thus γ_1,Fqu(χ)  qu−1. Denote by z = γ_1,Fqu(χ) and assume

thatz  qu−1− 1. Bythe definition of K-gonality, there is anon-constant morphism

w : χ→ P1 _{with deg(w)= z and definedoverF}

qu. Since w(χ(F_qu))⊆ P1(F_qu),we get

(χ(Fqu)) z(qu+ 1) (qu−1− 1)(qu+ 1),thatisacontradiction. 2

Remark 1. ByLemma 1, we can apply Theorem 2 to the Norm-Trace curve. In fact, wecanconsider thegonalitysequence overK of χ toget alowerbound onthe second generalizedHammingweightofthetwofamiliesofNorm-TraceLRCcodes:

• Lett 1 andletC bea(q2u−1_{,(t+ 1)(q}u−1_{− 1),q}u−1_{− 1) Norm-TraceLRCcode.} Thenwe have

d2(C) q2u−1+ qu−1− tqu−1− (qu−1− 1)(1 + q + · · · + qu−1).

• Let t  1 and let C be aNorm-Trace LRC code with parameters (q2u−1_{− q}u−1_, (t+ 1)(q +· · · + qu−1), q +· · · + qu−1).Thenwehave

5. HermitianLRCcodes

InthissectionweimprovetheboundonthedistanceofHermitianLRCcodesproposed in [1] using some properties of Hermitian codes which are a special case of algebraic geometric codes.

5.1. Hermitiancodes

Let us consider K = Fq2 afinite fieldwith q2 elements. The Hermitian curve H is

definedoverK bytheaffineequation

xq+1= yq+ y where x, y∈ K. (2) This curvehasgenusg = q(q₂−1) andhasq3_{+ 1 pointsofdegreeone,namelyapoleQ}

∞

and n= q3 rationalaffinepoints,denotedbyP_H={P1,. . . ,Pn}[31].

Definition 4.Let m∈ N suchthat0 m q3_{+ q}2_{− q − 2. Thenthe Hermitiancode}

C(m,q) isthecodeC(D,mQ_∞) where

D = 

αq+1_=βq_+β

Pα,β

is the sum of all places of degree one (except Q_∞, that is a point at infinity) of the Hermitian functionfieldK(H).

ByLemma 6.4.4.of [33]wehavethat

Bm,q={xiyj | qi + (q + 1)j  m, 0  i  q2− 1, 0  j  q − 1},

forms abasisofL(mQ_∞).Forthisreason,theHermitiancodeC(m,q) couldbe seenas Span_F

q2evPH(Bm,q) .Moreover,thedualofC(m,q) denotedbyC(m⊥,q)= C

⊥_{(m,q) is}

againanHermitiancodeanditiswellknown(Proposition 8.3.2of[33])thatthedegree

m of thedivisorhasthefollowingrelationwith respecttom_⊥:

m_⊥= n + 2g− 2 − m. (3) The Hermitian codes can be divided in four phases [11], any of them having specific explicit formulas linking their dimension and their distance [22]. In particular we are interestedinthefirstandthelastphaseofHermitiancodes,whichare:

I Phase: 0 m_⊥ q2_{− 2. Then we have m}

⊥ = aq + b where0 b a  q − 1 and



d = a + 1 if a > b

d = a + 2 if a = b. (4)

IV Phase: n− 1  m_⊥ n + 2g − 2. Inthiscasem_⊥= n+ 2g− 2− aq − b wherea,b

areintegerssuchthat0 b a q − 2 andthedistanceis

d = n− aq − b. (5)

5.2. Bound ondistanceof HermitianLRCcodes

LetK =Fq2 be afinite field,whereq is apowerofaprime.LetX = H bethe

Her-mitiancurvewithaffineequationas in(2).Werecallthatthiscurvehasq3_F

q2-rational

affinepointsplusoneat infinity,thatwedenotedbyQ_∞.

Weconsider two of the three constructionsof Hermitian LRCcodes proposedin[1]

and we improve the bound on distance of Hermitian LRC codes using properties of Hermitiancodes.Inparticular, ifwefind anHermitiancodeC(m,q)= CHer suchthat

CLRC ⊂ CHer,thenwehavedLRC  dHer.

Projection on x ByProposition 4 of [1],we havea familyof (n,k,r) Hermitian LRC codes with r = q− 1, length n = q3_{, dimension k = (t− 1)(q − 1) and distance d }

n−tq−(q−2)(q+1).Moreover,forthesecodes,S = K,D = tQ_∞forsome1 t q2₋₁ andthebasisforthevectorspaceV is

B = {xj_yi_{| j = 0, . . . , t, i = 0, . . . , q − 2}. (6)}

UsingtheHermitiancodes,weimprovetheboundonthedistanceforanyintegert,such thatq2_{− q + 1 t q}2_{− 1.}

To find an Hermitian code C(m,q) = CHer such that CLRC ⊂ CHer, we have to

computethe set Bm,q,that is,we have to findm. After that, to computethe distance

ofC(m,q) weuse(4)and(5). Weconsider thefirstHermitian phase:0 m_⊥ q2_{− 2,} thatis, q2_{− q + 1 t q}2_{− 1.}

Forthisphasem_⊥= aq +b,where0 b a q−1 andthedistanceoftheHermitian code is either d = a+ 1 if a > b or d = a+ 2 if a = b. By (6), m must be equalto

m= qt+ (q + 1)(q− 2) andby(3) wehavethatm_⊥= n+ 2g− 2− m= q(q2_{− t).So}

b= 0 anda= q2_{− t andthedistanceoftheHermitiancodeisd}

Her = a+ 1= q2− t+ 1,

sincea> b.Thisimpliesthat

dLRC  q2− t + 1, for any t  q2− q + 1. (7)

Notethat(7)improvestheboundonthedistanceproposedinProposition 4of[1]since

q2− t + 1 > q3− tq − (q − 2)(q + 1) ⇐⇒ t(q − 1) > q(q − 1)2+ 1 ⇐⇒ t > q2− q. Wejustprovedthefollowing:

Proposition 3.Letq2− q + 1 t q2− 1.Itispossibletoconstructafamily of(n,k,r) Hermitian LRCcodes{Ct}q2_−q+1tq2₋₁ withthefollowingparameters:

n = q3, k = (t− 1)(q − 1), r = q − 1 and d  q2− t + 1.

Two recoveringsets In[1]theauthors proposeanHermitiancodewithtworecovering sets ofsizer1= q− 1 andr2= q,denotedbyLRC(2). Theyconsider

L = Span{xiyj, i = 0, . . . , q− 2, j = 0, . . . , q − 1}

and a linear code C obtained by evaluating the functions in L at the points of B = g−1(Fq2\M),where g(x,y)= x and M = {a ∈ F_q | aq+ a = 0}. So|B| = q3− q. By

Proposition 4.3of[1],theLRC(2)codehaslengthn= (q2_{− 1)q,dimensionk = (q− 1)q} and distance

d (q + 1)(q2− 3q + 3) = q3− 2q2+ 3. (8) As before, we improve thebound on thedistance using Hermitian codes thatcontains the LRC(2) code.To dothis wehave to findm_⊥. ByL, we havethatm= q(q− 1)+ (q + 1)(q− 2) soweareinthefourthphaseofHermitiancodes becausem_⊥= n+ 2g− 2− m= q3_{− q}2_{+ q.Inthiscased}

Her = m⊥− 2g + 2= q3+ 2q + 2.Since |B|= q3− q,

we havethat

dLRC  dHer− q = q3+ q + 2. (9)

Note thatthisbound improvesbound(8).Wejustprovedthefollowingproposition: Proposition 4. LetC be a linear code obtained by evaluating the functions in L at the points ofB.ThenC hasthefollowingparameters:

n = (q2− 1)q, k = (q − 1)q, r1= q− 1, r2= q and d q3+ q + 2. Acknowledgment

Theauthors wouldlike tothanktheanonymousrefereesfortheircomments. References

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