Algebraic-geometric codes from vector bundles and their decoding

Algebraic-ge ometric codes from v ector b undles and their decoding V alentin Savin, CEA-LETI, MIN A TEC, Grenoble, Franc e, valentin.savin@cea.fr Abstract — Algebraic-geometric codes can be constructed by ev aluating a certa in set of functions on a set of distinct rational points of an algebraic cur ve. The set of functions that ar e ev aluated is the linear space of a gi ven divisor or , equ iv alently , the set o f section of a given line bundle. Using arbitrary rank vector bundles on algebraic curves, we propose a natural generalization of the abov e construction. Our codes can also be seen as interleav ed versions of classical algebraic-ge ometric codes. W e show that the algorithm of Brown, Minder and Sh okrollahi can be extended to thi s new class of codes and it corrects any n umber of errors up to t ∗ − g / 2 , where t ∗ is the designed corr ection capacity of the code and g is the curve genus. I . I N T RO D U C T I O N The construction of error correcting codes using methods from algebraic-g eometry was first pr oposed b y Gop pa [4], [5] in the ear ly ’80s. He constructed c odes by e valuating a certain set of d ifferential form s o n a set of distinct r ational po ints of an algebr aic curve. From a dual p oint of view , Goppa’ s cod es can be constructe d by evaluating a certain set of function s on a set of distinct ration al poin ts of an algeb raic cu rve [1]. The set of fun ctions that are ev a luated is the linear space of a gi ven divisor , whose su pport is disjoint fr om the set of ev a luation points. These codes generalize the Bose-Chaudhuri- Hocquen ghem ( BCH), Reed-Solo mon (RS) and Go ppa cod es (latest code s being introduc ed b y Gopp a in the ear ly ’70s). Unlike the RS codes, algebr aic-geometr ic (AG) co des are not generally MDS co des, but their Singleton d efect is upp er bound ed by the genus of the curve. Ho we ver , despite th eir Singleton defect, A G codes are b etter than RS co des since they allow the constru ction of lon ger codes over the same alphabet. Ano ther advantage of A G codes is that for fixed code par ameters, their enco ding and de coding algorithms run faster as th ey can be per formed in a smaller field . Sin ce the end of the 80s, intensive r esearch has be en don e on decod ing algorithm s and most of the meth ods used to deco de BCH, RS or Gopp a codes were exten ded to the class of AG co des [7]. In order to explain o ur approach to construct codes f rom vector bundles on pr ojective curves, let us r ecall the classical construction o f AG codes. Let C b e an absolu tely irredu cible, smooth, pr ojective curve of genus g , defin ed over a fin ite b ase field F q . Let P = { P 1 , P 2 , . . . , P n } be a set o f distinct ratio nal points of C and D ∈ Div ( C ) be a di visor wh ose support is disjoint from the set P . The linear code C ( P , D ) is d efined as the image of the ev aluation map: ev : L ( D ) − → F n q f 7− → ( f ( P 1 ) , f ( P 2 ) , . . . , f ( P n )) (1) where L ( D ) is the linear space of D . The parame ters of the code, or boun d on them, can be determined u sing well- known statemen ts in algebraic-geo metry , notably the Hasse- W eil theor em and the Rieman n-Roch theo rem, and it can be seen that the Singleton defect o f the code is upper boun ded by the cu rve genus g . Interleaved AG codes were defined in [3] as follows. Sup- pose that Q = q r and identify F Q and F r q as F q -vector spac es by fixing a basis of F Q over F q . For any p oint P ∈ C and any f = ( f 1 , f 2 , . . . , f r ) ∈ L ( D ) ⊕ r , the ev aluation vector f ( P ) = ( f 1 ( P ) , f 2 ( P ) , . . . , f r ( P )) ∈ F r q can be identified with an element o f F Q . T he cod e C ( P , D , r ) is defined as th e image of th e ev aluation m ap ev : L ( D ) ⊕ r − → F n Q f 7− → ( f ( P 1 ) , f ( P 2 ) , . . . , f ( P n )) (2) This cod e does not gen erally be F Q linear, howev er it is a F q -vector subspace of F n Q . T o explain our app roach, we first remar k tha t there is a one- to-one corr esponden ce between th e divisor c lass gr oup Cl ( C ) (the grou p of equiv alence classes of Div ( C ) ) and isomo rphism classes of line bundles on C . For any divisor D , we denote by O ( D ) the line bundle associated with D . The lin ear sp ace L ( D ) is isomorp hic to the spac e of glob al section s o f O ( D ) , which is g enerally deno ted b y H 0 ( C , O ( D )) . W e construct A G codes by ev aluating global sections o f arbitrary ran k vector bundles E o n th e po ints P 1 , P 2 , . . . , P n . Thus, the classical con struction of AG co des (1) cor respond s to the case E = O ( D ) an d the co nstruction of inter leav ed AG codes (2) corresponds to E = O ( D ) ⊕ r . In general, if E is a r ank- r vector bundle, we o btain a co de C ( P , E ) over F Q , not necessarily linear, where Q = q r . Further, we in vestigate the para meters of these cod es, or b ound on th em, such as the cod e leng th, dimension , an d minim um distance. While the code dimen sion can still be lower bo unded by th e Riemann - Roch theorem, it is mu ch hard er to compute its exact value or to g i ve a lower bou nd o n the minimu m distance. The main reason is that unlike line vector bundles, arb itrary rank vector bundles of negati ve d egree m ay have non zero sections. T o overcome such situa tions, we need the vector bundle to satisfy some stability con dition. When this condition is satisfied, we can compute the co de dimension and we can show that the Singleton defect is upper boun ded by the curve genus g . Now we come to the deco ding pr oblem. Bleichen bacher, Kiayias an d Y u ng [2] pr oposed a new d ecoding algo rithm for interleaved Reed-Solo mon codes over the Q -ary symmetr ic channel, which was la ter extended by Bro wn, Minder an d Shokro llahi to the case of inter leav ed AG cod es [3]. One advantage o f using interleav ed A G cod es is that they allow transmissions at rates closer to the ch annel ca pacity . If Q = q r = 2 hr , the Q -ary symmetric chan nel m odel app lies to settings wh ere packets of hr b its are sen t and error s are assumed to be bursty . From the coding theo ry perspective, errors on bits of the same packet are assum ed to b e co rrelated. This usually arri ves when packets of bits a re sent o ver different transmission cha nnels, some of wh ich may induce erro rs. When Q is too large, ef ficient decoding of codes designed over F Q is imp ossible, which explains the advantage of inter leav ed codes, since their d ecoding algorithms opera te over F q . In this paper we show that the deco ding algo rithm o f interleaved A G codes can be extended to the c lass of codes constructed from vector bundles and it correc ts any nu mber of erro rs u p to g / 2 from half the design ed m inimum distance o f the co de. In the n ext section we revie w some of the mathematical backgr ound need ed to understand the construction of A G codes from vector bundles on algebr aic curves. Ou r aim is n ot to do an exhaustiv e nor a self-con tained pr esentation, b u t rather to guide the reader from elemen tary to more complica ted objects and show that objects as vector b u ndles or stable vector bundles n aturally arise in algebr aic geo metry . In section s III an d IV we present respectively the con struction of A G codes fro m vector bundles and their decoding over the Q -ary symmetric chann el. Section V f ocuses o n the con struction of vector bundles verifying the stability condition. Finally , section VI con cludes this pap er . I I . V E C T O R B U N D L E S O N P RO J E C T I V E C U RV E S For more details on this topic we refer to classical algebraic- geometry texts, such as [ 6]. W e assume th at the reader is familiar with classical algebraic -geometr ic codes and basics on algeb raic-geom etry , such as cu rves an d divisors. W e first deal with algebraic closed fields, then we generalize to arbitrary fields by extending scalars to their algebraic closure. Let C be an irredu cible, smooth , projective cur ve o f genus g , over an algebraic closed field k (althoug h these co n- ditions a re not always nec essary). There are two po ssibilities of intro ducing vector bundles over k : they can be defined as locally fr ee she av es of finite rank over the cur ve C , or as families o f vector spaces p arameterized by C . While the first d efinition is often mo re conv enient for d eeper analy sis and un derstandin g, the secon d definition ha s the ad vantage of being more intu iti ve and compr ehensible to th ose not familiar with h eavy algebraic- geometry for malism. W e will f ocus on intuition and will introd uce vector bundles as families of vector spaces parameter ized by C . Precisely , a rank- r vector bundle over C is a variety E together with a surjective regular map E π → C , such that: 1) for any P ∈ C the fiber E P := π − 1 ( P ) is endowed with a structu re o f k -vector space of dim ension r , 2) for any P ∈ C , th ere is an open subset U ∈ C co ntaining P and a map ϕ : U × k r → π − 1 ( U ) such that: – ϕ ( P ′ , v ) ∈ E P ′ for any P ′ ∈ U – the restriction ϕ : { P ′ } × k r → E P ′ is a vector bundles isomorph ism for any P ′ ∈ U It may be co n venient to v isualize the second cond ition using the following co mmutative d iagram: U × F r q ϕ / / pr U A A A A A A π − 1 ( U ) =: E U π y y s s s s s s s s U One may thin k of a vector bundle as a family of vector spaces { E P } P ∈C parameteriz ed b y C , which looks locally trivial. Any algebraic operation with vector spaces can b e extend ed to vector bundles: for instance, we can define dir ect sums, tensor produ cts, exterior (wed ge) produ cts and du al vector bundles. A (globa l) section of the vector bundle E over C is a regular map s : C → E , such that s ( P ) ∈ E P for any P ∈ C . The set o f g lobal sections o f E is de noted by H 0 ( C , E ) . It is canonically en dowed with a k -vector space structure and its dimension is d enoted by h 0 ( C , E ) . A line bundle L over C is simply a r ank- 1 vector bundle. T o any mero morphic section s : C → L we associate a divisor ( s ) = Z ( s ) − P ( s ) , where Z ( s ) and P ( s ) denote respectively the set o f zero s a nd the set of p oles, counted with multiplicities. Note th at if s ∈ H 0 ( C , L ) is a g lobal section, then ( s ) = Z ( s ) is an effective divisor . Th e degr ee of L is by definition the degree of ( s ) . deg( L ) := deg ( s ) The fact that deg( L ) is well d efined f ollows from the fir st assertion of the followi ng proposition. The second assertion highligh ts the connection between g lobal sections of line bundles and linear spaces. Pr op osition 1: ( a ) If s , s ′ are meromorph ic sectio ns o f a line bundle L , then ( s ) an d ( s ′ ) are linear equ i valent divisors. ( b ) For any mero morph ic sectio n s , th e map L ( s ) − → H 0 ( C , L ) f 7− → f s defines an isomo rphism of vector sp aces. If E is a ran k- r vector bundle o n C , its r -th exterior power det( E ) := ∧ r E is a line bundle, called the determinant b undle of E . Th e degr ee o f E is b y de finition the degree of its determinan t bundle: deg( E ) := deg(det ( E )) The slope o f E is d efined by µ ( E ) = deg( E ) rank ( E ) . Examples of vector bundles that natur ally arise in algebraic - geometry are the tangent bundle and its du al, called the cotangen t bundle, o r if the curve C is emb edded in some projective space, the normal bundle. The cotan gent bundle of C is also called the canonical bundle and is denoted by Ω . Any divisor associated with the cano nical bundle is called cano nical d ivisor . W e can n ow state the Riemann-Roch theorem [6]. Theor em 2 (R iemann-R och): Let E be a vector b undle of rank r an d degree e o n C . Then: h 0 ( C , E ) − h 0 ( C , Ω ⊗ E ∗ ) = e + r (1 − g ) where E ∗ is the dual vector bundle of E . At this po int we have introdu ced the necessary too ls f or defining A G codes from vector bundles. In ord er to be able to in vestigate the par ameters o f these co des and their decoding algorithm , we n eed th e vector bundles to satisfy some stability condition . The theory of stable vector bundles goes back to the classification prob lem o f vector bundles in the 6 0s. Howe ver , in this pape r we need only a we aker version of the stability , which we call weak stab ility . Befo re introd ucing the stability condition , we have to be more specific a bout morphisms o f vector bundles an d vector sub-bundles. Let E and F be two vector bundles on C . A morphism of vector bundles is a regular m ap ϕ : E → F , such th at: • for any P ∈ C an d any x ∈ E P , ϕ ( x ) ∈ F P , • for a ny P ∈ C th e induced map ϕ P : E P → F P is a morph ism of vector spac es. Any morphism of vector bundles ϕ : E → F induces in a obvious way a mo rphism between the co rrespon ding vecto r spaces of global section s, that is ϕ : H 0 ( C, E ) → H 0 ( C, F ) . W e say that E is a sub -bundle of F if there is a morph ism of vector bundles ϕ : E ֒ → F , such that for any po int P ∈ C the induced morph ism ϕ P : E P → F P is injectiv e. In this case one can define a q uotient vecto r bundle F /E , wh ose fiber in a po int P ∈ C is d efined by ( E / F ) P = E P /F P . Definition 3: A vector bundle E is said to be weakly stab le if for any line sub-bundle L ⊂ E the following inequality holds: deg( L ) ≤ µ ( E ) Pr op osition 4: ( a ) Any line bundle is weakly stable. ( b ) L et E = n ⊕ i =1 E i be a direct su m of vector bundles. T hen E is weakly stable if and only if µ ( E 1 ) = · · · = µ ( E n ) and all E i , i = 1 , . . . , n , are weakly stab le. ( c ) I f E is weakly stable then for any line bundle L , the tensor produ ct E ⊗ L is also w eakly stable. Pr op osition 5: Assume th at E is a weakly stable vector bundle on C . ( a ) Any global s ection of E vanishes in at most ⌊ µ ( E ) ⌋ poin ts. ( b ) If deg( E ) < 0 then h 0 ( C , E ) = 0 . I I I . A G C O D E S F RO M V E C T O R B U N D L E S In this section we define algebraic-geom etric codes from vector bundles. Thro ugh the r est o f this paper, we denote by C a n absolutely irreducible, smooth, projective curve of genus g , defined over a finite base field F q . For any vector bundle E → C , let ¯ E → ¯ C be the vector bundle o btained by exten ding scalars fro m F q to its algebr aic closur e ¯ F q . W e define deg( E ) = deg( ¯ E ) an d we say that E is weakly stable iff ¯ E is weakly stab le. Let P = { P 1 , . . . , P n } b e a set of distinct rational points of C , E be a rank- r vector bundle on C , and set Q = q r . W e fix once for all basis of F Q and E P i , i = 1 , . . . , n , as vector spaces over F q , which allows u s to identify E P i ≃ F Q ≃ F r q . Henceforth, th ese identifications will be used withou t recalling the subjac ent basis. The alg ebraic- geometric code C ( P , E ) over F Q is defined as the im age of the ev aluation m ap: ev : H 0 ( C , E ) − → n ⊕ i =1 E P i ≃ F n Q f 7− → ( f ( P 1 ) , f ( P 2 ) , . . . , f ( P n )) (3) Note that th is is not n ecessarily a F Q linear code, but it is a F q linear sub space of F n Q . T he length of the c ode is n a nd fo r the other p arameters, the fo llowing notation s will be used: • K is the size of the code. • k is the dimension of th e co de; since it is no t necessarily a linear code its d imension is defin ed by: k = lo g Q K ∈ R • d is the minim al distance of the c ode. For arbitrary non-lin ear codes, d cor respond s to the minimal distanc e between any two co dew ords. However , since C ( P , E ) is F q linear, d is also eq ual to the minimal weight of a non-ze ro codeword. Note th at if the e valuation map is injectiv e, then K = q h 0 ( C ,E ) and there fore k = h 0 ( C , E ) r . Theor em 6: Assume that E is a weakly stable vector b u ndle of degree e a nd slope µ = e /r < n . Then: ( a ) the ev aluation ma p is injective, ( b ) d ≥ n − ⌊ µ ⌋ , ( c ) k ≥ µ + 1 − g , ( d ) the Singleton d efect of the co de is uperb ound ed by the curve g enus g . Pr oo f . Since E is weakly stable, any section f ∈ H 0 ( C , E ) vanishes in at most ⌊ µ ⌋ points, which proves ( a ) and ( b ) . Because th e ev aluation map is injective ( a ) , we also have k = h 0 ( C , E ) /r . By the Riemann- Roch th eorem h 0 ( C , E ) ≥ e + r (1 − g ) , there fore k ≥ µ + 1 − g . Finally , ( d ) follows from ( b ) an d ( c ) .  Pr op osition 7: Let E be vector bundle o f degree e and slope µ = e/ r . Assum e that bo th E and E ∗ are weak ly stable an d µ > 2 g − 2 . Then k = µ + 1 − g . Pr oo f. The ten sor produ ct Ω ⊗ E ∗ is a weakly stable vector bundle of degree: deg(Ω ⊗ E ∗ ) = r deg(Ω) − deg( E ) = r (2 g − 2 ) − e < 0 Therefo re h 0 (Ω ⊗ E ∗ ) = 0 and the assertion f ollows by the Riemann-Roch theo rem.  I V . D E C O D I N G A L G O R I T H M Let C ( P , E ) be a code ov er F Q defined by a ran k- r vector bundle E an d an evaluation set P = { P 1 , . . . , P n } . Assume that the code word ( f ( P 1 ) , f ( P 2 ) , . . . , f ( P n )) , defined by some f ∈ H 0 ( C , E ) , is transmitted over the Q - ary symmetric c hannel and let ( y 1 , y 2 , . . . , y n ) be the re ceiv ed word. Our goal is to decode th e codeword and fo r this we proceed in a way similar to [3]. Let t be a pa rameter to be determined latter an d let L b e a li ne b undle of degre e l := t + g . The deco ding works in two steps as follows: ( S 1) Find a non- zero elemen t ( v , w ) ∈ H 0 ( C , E ⊗ L ) × H 0 ( C , L ) such that v ( P i ) = y i ⊗ w ( P i ) , ∀ i = 1 , . . . , n If ( v , w ) d oes n ot exist, o utput a d ecoding erro r . ( S 2) I f ther e exists ¯ f ∈ H 0 ( C , E ) su ch that v = ¯ f ⊗ w decode ¯ f . Otherwise, ou tput a deco ding error . Pr op osition 8: Let ǫ den ote the num ber of err ors incu rred during tr ansmission. If ǫ ≤ t then th ere exists an non- zero element ( v , w ) satisfying ( S 1) . Pr oo f . Assum e that erro rs o ccur in po ints P i 1 , . . . , P i ǫ and let D err = P i 1 + · · · + P i ǫ . Using the Riemann-Roch theorem it can be proved that h 0 ( C , L ( − D err )) > 0 . Therefor e, we can choose a n on-zer o w ∈ H 0 ( C , L ( − D err )) and define v = f ⊗ w . If P i is not an er ror point, meaning that P i 6∈ { P i 1 , . . . , P i ǫ } , then y i = f ( P i ) and so v ( P i ) = y i ⊗ w ( P i ) . Otherwise, the equ ality v ( P i ) = y i ⊗ w ( P i ) still h olds, be cause v ( P i ) = w ( P i ) = 0 for any P i ∈ { P i 1 , . . . , P i ǫ } .  Theor em 9: Let ǫ den ote th e numb er o f error s incurred during tran smission. Assume tha t E is a weak ly stable vector bundle of degree e and slope µ = e/r , such that: ǫ ≤ t and ǫ + t ≤ n − µ − g Then the a bove decod er o utputs the tra nsmitted codeword. Pr oo f. From the above prop osition, th ere exists a non-ze ro element ( v , w ) ∈ H 0 ( C , E ⊗ L ) × H 0 ( C , L ) satisfying ( S 1) , that is: v ( P i ) = y i ⊗ w ( P i ) , ∀ i = 1 , . . . , n Assume that err ors occur in points P = { P i 1 , . . . , P i ǫ } a nd let D = P P i 6∈P P i . For any P i 6∈ P we have y i = f ( P i ) and therefor e: ( v − f ⊗ w )( P i ) = y i ⊗ w ( P i ) − f ( P i ) ⊗ w ( P i ) = 0 It f ollows that v − f ⊗ w ∈ H 0 ( C , E ⊗ L ( − D )) . On the other h and, kn owing that deg( E ) = e , deg( L ) = t + g a nd deg( D ) = n − ǫ , we get: deg( E ⊗ L ( − D )) = e + r ( t + g − ( n − ǫ )) = r ( ǫ + t − n + µ + g ) < 0 Consequently , E ⊗ L ( − D ) is a weak ly stable vector bundle of negativ e degree, so it has no non- zero glo bal section s. Hence v = f ⊗ w and the decod er o utputs f .  Note that the d esigned cor rection capac ity of the c ode f or the Q -ar y symmetric chan nel is t ∗ =  n − µ 2  . From the above theor em, it fo llows that the decodin g algor ithm c orrects any p attern of ǫ < t ∗ − g 2 errors. W e note that the above algorithm can easily be extended to correct both errors and erasures. Throu ghout the rest of th is sectio n we gi ve a possible realization the decodin g algorith m. Our goa l is ju st to pr ove that th e decodin g algorithm is c onstructible and executable in polyno mial time. W e fix once f or all: • a basis of F Q over F q , • a basis of E P over F q , fo r each P ∈ P , • a basis of L P over F q , fo r each P ∈ P , • f 1 , . . . , f h a basis of H 0 ( C , E ) over F q , • ϕ 1 , . . . , ϕ a a basis of H 0 ( C , E ⊗ L ) over F q , • ψ 1 , . . . , ψ b a basis of H 0 ( C , L ) over F q The first th ree basis allow us to identify : E P ⊗ L P ≃ E P ≃ F Q ≃ F r q Let ( y 1 , y 2 , . . . , y n ) b e the received w ord. For each 1 ≤ i ≤ n , we con sider th at y i ∈ F Q ≃    y i, 1 . . . y i,r    ∈ F r q and we set Y =      y 1 y 2 . . . y n      ∈ M nr,n ( F q ) where each y i is iden tified with the correspo nding column vector . Moreover , we define: F P i = ( f 1 ( P i ) , f 2 ( P i ) , . . . , f h ( P i )) ∈ M 1 ,h ( F Q ) ≃ M r,h ( F q ) F =      F P 1 F P 2 . . . F P n      ∈ M n,nh ( F Q ) ≃ M nr,nh ( F q ) V =      ϕ 1 ( P 1 ) ϕ 2 ( P 1 ) · · · ϕ a ( P 1 ) ϕ 1 ( P 2 ) ϕ 2 ( P 2 ) · · · ϕ a ( P 2 ) . . . . . . . . . . . . ϕ 1 ( P n ) ϕ 2 ( P n ) · · · ϕ a ( P n )      ∈ M n,a ( F Q ) ≃ M nr,a ( F q ) where all f j ( P i ) and ϕ j ( P i ) a re identified with r × 1 colu mn vectors in F r q , and W =      ψ 1 ( P 1 ) ψ 2 ( P 1 ) · · · ψ b ( P 1 ) ψ 1 ( P 2 ) ψ 2 ( P 2 ) · · · ψ b ( P 2 ) . . . . . . . . . . . . ψ 1 ( P n ) ψ 2 ( P n ) · · · ψ b ( P n )      ∈ M n,b ( F q ) The deco ding alg orithm can now be d escribed as fo llows: ( S 1) Find a no n-zero solutio n ( v 1 , . . . , v a , w 1 , . . . , w b ) ∈ F a + b q of the system V    v 1 . . . v a    = Y W    w 1 . . . w b    If the system does n ot h av e a non- zero solu tion, ou tput a deco ding er ror . ( S 2) Find a solution λ = ( λ 1 , . . . , λ h ) ∈ F h q of the system V    v 1 . . . v a    = F Λ W    w 1 . . . w b    where Λ =    t λ . . . t λ    ∈ M nh,n ( F q ) and output ¯ f = λ 1 f 1 + · · · + λ h f h . If the system does not have any solution, output a decoding error . Note tha t th e second step o f the ab ove realiza tion comp ute a section ¯ f = λ 1 f 1 + · · · + λ h f h verifying v ( P i ) = ¯ f ( P i ) ⊗ w ( P i ) , ∀ i = 1 , . . . , n where v = v 1 ϕ 1 + · · · + v a ϕ a and w = w 1 ψ 1 + · · · + w b ψ b . This is a little bit dif ferent from th e secon d step of the d ecoding algorithm , which requ ires the above equ ality to hold fo r any point P ( i.e. v = ¯ f ⊗ w ). Assumin g that the vector bundle E is weak ly stable, any no n-zero section of E ⊗ L vanishes in at most µ ( E ⊗ L ) = µ + t + g points. Fu rthermo re, assumin g that n > µ + t + g and v ( P i ) = ¯ f ( P i ) ⊗ w ( P i ) , ∀ i = 1 , . . . , n , it fo llows that th e section v − ¯ f ⊗ w v anishes in m ore than µ + t + g p oints, an d so v = ¯ f ⊗ w . V . C O N S T RU C T I O N O F W E A K LY S TA B L E V E C T O R B U N D L E S Most of statements conce rning the param eters and the decodin g of algebraic codes constructed fro m vector bundles require weakly stable vector bundles. A tri v ial exam ple of a weakly stable vecto r bundle o f ran k r and degree e is given by the direct sum of r line bundles of degre e e . Such a vector bundle is called com pletely und ecompo sable. In this section we show that for any cur ve o f genus g ≥ 2 a nd any in tegers r > 0 a nd e ther e exist no n-trivial examples of weak ly stable vector bundles of rank r and d egree e . Let e = αr + β , with α, β ∈ Z such that 0 ≤ β < r . Let F 1 , F 2 and F be line bundles o n C , such that: deg( F 1 ) = deg( F 2 ) = α, deg( F ) = α + 1 Consider a sequence of vecto r bundles E i defined by the following no n-trivial extensions: E 1 = F 1 0 → E 1 − → E 2 − → F 2 → 0 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 0 → E r − β − 1 − → E r − β − → F 2 → 0 0 → E r − β − → E r − β +1 − → F → 0 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 0 → E r − 1 − → E r − → F → 0 The exten sions of F 2 by E i , i ≤ r − β − 1 , ar e classified by H 1 ( C , F ∗ 2 ⊗ E i ) ≃ H 0 ( C , Ω ⊗ F 2 ⊗ E ∗ i ) , by Poinc ar ´ e duality . Since F ∗ 2 ⊗ E i is a vector bundle of degree 0 and ran k i , by th e Riemann-Roch theo rem we obtain h 0 ( C , Ω ⊗ F 2 ⊗ E ∗ i ) > 0 . It fo llows that there exist n on-trivial extensions o f F 2 by E i . Using similar argum ents, it can also b e shown tha t there exist non-tr i vial extensions of F by E i , i ≥ r − β . Pr op osition 10: Le t E := E r . The n: ( i ) deg( E ) = αr + β = e and r k ( E ) = r . ( ii ) E is weak ly-stable. Pr oo f. ( i ) is clear f rom construc tion. T o see that E is weakly stable, consider L a line sub-bundle of E and let i b e th e smallest index such th at L is contained in E i . Th en th e morph ism L → E i → E i /E i − 1 is non zero, therefo re: deg( L ) ≤ deg ( E i /E i − 1 ) ≤ α + 1 If deg( L ) = α + 1 then L ≃ E i /E i − 1 and the extension of E i /E i − 1 by E i would split. So deg ( L ) ≤ α ≤ µ ( E ) , wh ich proves that E is weakly stab le.  V I . C O N C L U S I O N S A ne w construction o f A G codes from vector bundles on algebraic curves was p roposed in th is paper, which allows a unified tr eatment of classical AG cod es and more recen tly interleaved A G co des. In the same time, this constru ction extends the ab ove class of AG c odes to a m uch larger class of codes. These new codes have very go od p roperties and they can b e d esigned over very large Galois fields with reasonable decodin g comp lexity , since decoding can be p erform ed in a smaller field. W e h av e also provid ed a deco ding algo rithm for these codes that c orrects any number o f errors up to t ∗ − g / 2 , where t ∗ is the design ed co rrection capacity of the co de and g is the cur ve g enus. The aim of this paper is also to relate the co nstruction of AG codes to mo re sophisticated an d p owerful concep ts in a lgebraic-g eometry . Howe ver , this is only a first step and more work has to be don e in this area. It is very likely that for suitable cho ices o f vector bundles E and L , th e deco ding algorithm will correct errors up to the designed correc tion capacity of the code. W e th ink that future work could bring out many useful in teractions between algebraic geometric codes and vector bundles on alge braic curves. R E F E R E N C E S [1] I. Blake , C. Heegard, T . Høholdt, and V . W ei. Algebra ic-geome try codes. IEEE T rans. Inform. Theory , 44(6):2596–261 8, 1998. [2] D . Bleiche nbache r , A. Kiayias, and M. Y ung. Decodi ng interlea ved Reed Solomon codes ove r noisy channels. In ICALP 2003, Procee dings of , pages 97–108, 2003. [3] A . Bro wn, L. Minder , and M. A. Shokrollahi. Impro v ed decoding of interl ea ved AG codes. Lectur e Notes in Computer Scienc e , 3796:37– 46, 2005. [4] V . D. Goppa. Codes on alg ebraic cu rves. Dokl. Acad. Nauk SSSR , 259:1289– 1290, 1981. Tran slation : Soviet Math. Dokl., vol. 24, pp. 170- 172, 1981. [5] V . D. Goppa. Algebraic-g eometric codes. Izv . A kad. Nauk SSSR , 46, 1982. Transl ation: Math. USSR Izv ., vol. 21, pp. 75-91, 1983. [6] R. Hartshorne. Algebr aic geometry . Graduate T exts in Mathematics. Springer -V erlag, 1977. [7] T . Høholdt and R. Pellika an. On the decoding of algebraic- geometri c codes. IEEE T rans. Inform. Theory , 41(6):1589–161 4, 1995.