Asymptotic Bound on Binary Self-Orthogonal Codes

1 Asymptotic Bound on Binary Self-Orthogonal Codes (confirmation no 18 333) Y ang Ding Abstract — W e present two constructions for binary self- orthogonal codes. It turns out that our constructions yield a constructive bound o n binary self-orthogonal codes. In particular , when the inf ormation rate R = 1 / 2 , by our co nstructive lower bound, the relativ e minimum distance δ ≈ 0 . 0595 (fo r GV b ound, δ ≈ 0 . 110 ). Mor eov er , w e have prov ed that th e bi nary self- orthogonal codes asymptotically achieve the Gilbert-V arshamo v bound. Index T erms — Algebraic geometry codes, concatenated codes, Gilbert-V ar shamov bound, Reed-Muller codes, self-dual basis, self-orthogonal codes. I . I N T R O D U C T I O N In cod ing theor y , we are in terested in good codes with lar ge length, i.e., we want to find a family of co des with length tending to ∞ . For a f a mily of linear [ n, k , d ] codes over F q , the ratio R := lim n →∞ k /n and δ := lim n →∞ d/n d enote the information rate and the re lati ve minimum distance, respectiv ely , of the codes. Th e set U q ⊆ [0 , 1] × [0 , 1 ] which is d efined as follows: a point ( δ, R ) ∈ R 2 with 0 ≤ δ ≤ 1 and 0 ≤ R ≤ 1 belo ngs to U q if an d only if there exists a sequence { C i = [ n i , k i , d i ] } i ≥ 0 of c odes over F q such th at n i → ∞ , d i n i → δ and k i n i → R , as i → ∞ . A main coding pr oblem is to determ ine the domain U q . Man in and Vl ˘ adut ¸ gave a descrip tion of U q throug h a fun ction α q : [0 , 1 ] → [0 , 1] which is defined by α q ( δ ) = sup { R : ( δ, R ) ∈ U q , for δ ∈ [0 , 1] } . It is well-known th at the function α q is continu ous an d decreasing, see [1]. An [ n, k ] linear cod e C over th e finite field F q is a lin ear k -dimen sional subspace of F n q . T he du al code C ⊥ of C is defined as th e orth ogona l space of C , i.e. , C ⊥ = { y ∈ F n q | xy = 0 fo r every x ∈ C } , where xy = x 1 y 1 + x 2 y 2 + · · · + x n y n is th e ordinary scalar produ ct of vectors x = ( x 1 , x 2 , · · · , x n ) , y = ( y 1 , y 2 , · · · , y n ) in F n q . A code C is self-othogona l if C ⊆ C ⊥ , and self- dual if C = C ⊥ . It is well-known that there exists a class of long bi- nary self-dua l codes which meet th e Gilbert-V arsh mov bo und The work of Y . Ding w as support ed by the China Sch olarship Council. Y . Ding is with the Department of Mathematics, Southea st Uni versity , Nan- jing, 210096, People’ s Republic of China (e-mail : PG23067461@n tu.edu.sg). The work of Y . Ding w as carrie d out while the author was studying in Di vision of Mathematic al Sc ience s, School of Physical and Mathematic al Science s, Nanya ng T echnological Uni versity , Singapore under the exchange program. [2]. W e emp loy the method which mentioned in [2], proof that binary self-orthog onal co des also achieve the Gilbert- V arshmov boun d. Howe ver , this result is n ot co nstructive. T o obtain the constructive bound on R and δ , we inv o lved two different ways to constru ct binary self-o rthogo nal codes. Both of the two con structions are based on a kind of alge- braic g eometry codes which achieves the Tsfasman- Vl ˇ adut ¸ - Zink bound. In the Construction A, we concatenate algebraic geometry codes with binary self-orthog onal co des to obtain th e desired codes. I n the Constru ction B, we also ge t the desired codes by consider ing s elf-orth ogonal algebraic geometry code s and express these alg ebraic geo metry co des into binary self - orthog onal cod es by employing the self-dual basis. Using these two constructio ns, we ob tain a lower bound on R an d δ . I n particular, u sing Constru ction B, we get δ ≈ 0 . 0595 when R = 1 / 2 (by Gilbert-V arshamov bo und, δ ≈ 0 . 110 when R = 1 / 2 ). This co rrespond ence is organized as follo ws. W e first recall some basic results o f concatenated codes, Reed- Muller c odes, Gilbert-V arsham ov bound, and some well-known facts abo ut algebraic geometry codes which are necessary for our purp ose. The main description of our two constructions are g i ven in Section I II, and we calcu late some examples. In Section IV , we hav e s hown th at there exists a binary self-orthogon al cod e achieving th e Gilbert-V a rshamov bou nd. The con clusion of this paper is given in the last section. I I . P R E L I M N A R I E S In this section, we giv e some fund amental properties about concatenate d codes, algebraic geometry codes and Reed- Muller codes. W e recall the results in [3], [4] and [5] as follows. Let C b e an [ s, v , w ] code over F q k and, fo r i = 1 , 2 , ..., s , let π i : F q k → F n i q be an F q -linear injective map whose im age C i = im( π i ) is an [ n i , k , d i ] cod e over F q . The imag e π ( C ) of th e following F q -linear injective map: π : C → F n 1 + ... + n s q (1) c = ( c 1 , ..., c s ) 7− → π ( c ) = ( π 1 ( c 1 ) , ..., π s ( c s )) is an [ n 1 + ... + n s, v k ] linear concatenated code over F q . From the d efinition of the concatenated c ode, we know that if two cod es C , C ′ over F q k satisfy C ⊆ C ′ , th en the two concatenate d codes π ( C ) ⊆ π ( C ′ ) over F q . Lemma 1 : If im( π i ) = [ n i , k , d i ] (1 ≤ i ≤ s ) are self- orthog onal codes, then π ( C ) is also a self-o rthogo nal code. Pr oof: Giv en any two codewords π ( c ) = ( π 1 ( c 1 ) , ..., π s ( c s )) and π ( c ′ ) = ( π 1 ( c ′ 1 ) , ..., π s ( c ′ s )) 2 of π ( C ) , where c = ( c 1 , c 2 , · · · , c s ) and c ′ = ( c ′ 1 , c ′ 2 , · · · , c ′ s ) are two codewords of C . Then ( π ( c ) , π ( c ′ )) = s X i =1 ( π i ( c i ) , π i ( c ′ i )) , where ( , ) stands f or the ord inary scalar produ ct o ver F q . Since im( π i ) (1 ≤ i ≤ s ) are self-orthog onal, we have π i ( c i ) · π i ( c ′ i ) = 0 for all 1 ≤ i ≤ s. Thu s π ( c ) · π ( c ′ ) = 0 , therefo re, π ( C ) is a self-or thogon al co de. Lemma 2 : ([3]) Suppose the images im( π i ) (1 ≤ i ≤ s ) are iden tical and have parameter s [ n, k , d ] . T hen π ( C ) is an [ ns, v k ] lin ear code over F q with the minimu m d istance at least w d . From now on , we assum e that the images im( π i ) 1 ≤ i ≤ s are identical, and de note as im( π ∗ ) , i.e., π (( c 1 , c 2 , · · · , c s )) = ( π ∗ ( c 1 ) , π ∗ ( c 2 ) , · · · , π ∗ ( c s )) in eq uation (1 ). Next, we review some basic con clusions of algeb raic ge om- etry c odes. Let X b e a smooth, p rojective, ab solutely irreducible cu rve of genus g define d over F q , let D be a set of N F q -rational points of X and let G be a n F q -rational divisor of X such that supp( G ) ∩ D = ∅ and 2 g − 2 < deg( G ) < N , where supp( G ) and deg( G ) d enote th e sup port and th e degree of G , respectively . Then the f unctional algebraic- geometry code C L ( G, D ) with param eters [ N , deg( G ) − g + 1 , N − deg( G )] can b e defined, see [1]. Let q = l 2 be a squar e. It is k nown that ther e exists a family of algebraic curves { X i } over F q with g i → ∞ attainin g the Drinfeld-V l ˇ adut ¸ bou nd, i.e., lim i →∞ sup( N ( X i / F q ) /g i ) = l − 1 where N ( X i / F q ) and g i are the number of F q -rational points and th e gen us of X i , respectively (see [4]). T hen, the paper [6] co nstructs a family of algeb raic geometry codes T i = C L ( G i , D i ) = [ N i , K i , D i ] q achieving the Tsfasman-Vl ˇ adut ¸ - Zink bo und wh ere D i contain all F q -rational p oint except only one ratio nal point P which is the sup port of d i visor G i , i.e. , we hav e R 1 + δ 1 = 1 − 1 l − 1 (2) where R 1 : = lim i →∞ K i N i and δ 1 : = lim i →∞ D i N i . denote the infor mation rate and the relative minimu m distance, respectively , o f the cod es. For the Con struction A, we also need some prop erties of Reed-Muller co des. Let v = ( v 1 , ..., v m ) denote a vector wh ich ra nges over F m 2 , and f is the vector of length 2 m by list of values which a re taken by a Boolean function f ( v 1 , ..., v m ) on F m 2 . Definition 1 : ([3]) Th e r th order binary Reed-Muller code (or RM co de) R ( r, m ) o f length n = 2 m , for 0 ≤ r ≤ m , is the set of all vectors f , wher e f ( v 1 , ..., v m ) is a Boo lean function which is a polyn omial of degree at most r. Lemma 3 : ([3]) Th e r th order binary Reed-Mu ller code R ( r , m ) has d imension k = P r i =0  m i  and minimum distance 2 m − r for 0 ≤ r ≤ m , where  m i  are binomial co efficients. Lemma 4 : ([3]) R ( m − r − 1 , m ) is the dual co de of R ( r , m ) with re spect to the ordin ary scalar pro duct, for 0 ≤ r ≤ m − 1 . From the d efinition of Reed-Muller codes, it is easy to known that we have R ( r 1 , m ) ⊆ R ( r 2 , m ) when 0 ≤ r 1 ≤ r 2 ≤ m. By Lem ma 4, wh en r ≤ ⌊ m − 1 2 ⌋ , R ( r, m ) is a self- orthog onal code. I n particular, when m is a n odd nu mber, R ( m − 1 2 , m ) is a self-d ual co de. Now , let m go thr ough all positive odd numbe r , we get a family of self-dua l Reed- Muller co des R ( r , m ) , where r = m − 1 2 , with param eters [2 m , 2 m − 1 , 2 m +1 2 ] . At the en d o f this section , in ord er to comp are our boun d with the existed b ound, w e g i ve the asymptotically Gilber t- V arshamov boun d. Lemma 5 : (Asymp totic Gilb ert-V arshamov Bound) If 0 ≤ δ ≤ q − 1 q then α q ( δ ) ≥ 1 − H q ( δ ) , (3) where H q ( δ ) is q -ary entropy fu nction de fined by H q ( x ) = ( x log q ( q − 1) − x log q x − (1 − x ) log q (1 − x ) , 0 < x ≤ ( q − 1) /q ; 0 , x = 0 . Remark 1 : In Fig.1 we sh ow this b ound f or q = 2 . I I I . C O N S T R U C T I O N S O F S E L F - O RT H O G O N A L C O D E S In this Section, we will presen t two construc tions of bin ary self-ortho gonal codes. A. Constructio n A Assume th at q = 2 2 t in this subsection. Let im( π ∗ ) be an binary [ n , 2 t ] linear co de. Let T i = [ N i , K i , D i ] be a family of alg ebraic g eometry codes over F 2 2 t achieving the T sfasman-Vl ˇ adut ¸ - Zink b ound, i.e., R 1 + δ 1 = 1 − 1 2 t − 1 (4) where R 1 : = lim i →∞ K i N i and δ 1 : = lim i →∞ D i N i . Now we state ou r first constru ction. Pr opo sition 1: L et C 0 be a self-orth ogonal code over F 2 with parameters [ n, 2 t, d ] , take C 0 as im( π ∗ ) , concaten ate the family of algebraic geometr y codes T i and C 0 under the map π = ( π ∗ , π ∗ , · · · , π ∗ ) , then we obtain a family of bina ry self-ortho gonal cod es C i with parameter s [ nN i , 2 tK i , dD i ] . Moreover , we have asymptotic eq uation R + 2 t d δ = 2 t n (1 − 1 2 t − 1 ) (5) where R and δ de note the informa tion rate and the relativ e minimum distance, respectively , of the con catenated co des C i . Pr oof: The r esult follow immediately consequen ce of the proper ties of algeb raic geometr y codes T i and concatenated codes. Now we gi ve som e examp les to illustrate th e r esult in Proposition 1. Example 1 : (RM cod es) If we fixed an o dd n umber m ≥ 3 , then we get a binary self-dual code [ 2 m , 2 m − 1 , 2 ( m +1) / 2 ] . Let 2 t = 2 m − 1 , then F 2 2 t = F 2 2 m − 1 . 3 T ABLE I E X A M P L E 2 binary codes t equ ations for R a nd δ [22 , 10 , 8] 5 R + 5 4 δ = 150 341 [24 , 12 , 8] 6 R + 3 2 δ = 31 63 [28 , 14 , 6] 7 R + 7 4 δ = 63 127 [40 , 20 , 8] 10 R + 5 2 δ = 511 1023 [44 , 22 , 8] 11 R + 11 4 δ = 1023 2047 [64 , 32 , 12] 16 R + 8 3 δ = 32767 65535 It is well-k nown that there exists a family of algebraic geometry co des T i over F 2 2 m − 1 with parameter s [ N i , K i , D i ] satisfy the equation (4). Then by Proposition 1, we get a f amily of binary concatenated codes C i = [2 m N i , 2 m − 1 K i , 2 m +1 2 D i ] . Asymptotically , we have the equation R + 2 m − 1 2 δ = 1 2 (1 − 1 2 2 m − 2 − 1 ) (6) Thus when we go throu gh all odd numb er m ≥ 3 , we get a sequence o f equations fo r R and δ . Example 2: (Some special binary self- orthogo nal cod es) From [ 9 ] and [ 7 ] , we get several optimal self-ortho gonal codes. Using these cod es to do the concatenation, we get some equations abou t R and δ for small t . W e list them in T ab le I. The la st column o f T able I was c alculated by (5). Using these two exam ples, we g et an asymptotic bo und for α 2 ( δ ) . B. Constructio n B In this subsection, we will give anoth er construction of binary self-o rthogon al codes. Let us first r ecall the defin ition of self -dual basis. Let { e 1 , · · · , e k } b e an F q -basis of F q k . A set { e ′ 1 , · · · , e ′ k } of F q k is called th e du al basis of { e 1 , · · · , e k } if we have T r F q k / F q ( e i e ′ j ) = δ ij =  0 , i 6 = j ; 1 , i = j , (Kronecker symbol). It is we ll-known tha t the dual basis always exists. W e say that a basis is self-d ual if it is its own dual. I t is well-kn own that the self-dual b asis alw a ys exists when c har( F q ) = 2 . Now we consider th e finite field F 2 2 t , we k now th at ther e exists a self-d ual F 2 -basis { e 1 , · · · , e 2 t } o f F 2 2 t . Th en for any element α in F 2 2 t , ther e exists a uniqu e 2 t -tup le vector α ( e ) = ( α 1 , · · · , α 2 t ) ∈ F 2 t 2 such that α = P 2 t i =1 α i e i . For any two elements α and β of F 2 2 t , we h av e T r F 2 2 t / F 2 ( αβ ) = ( α ( e ) , β ( e ) ) = 2 t X i =1 α i β i , where ( , ) stands for the ordinar y scalar pr oduct over F 2 . Thus, we h av e a one-to-o ne corresp ondence ρ between F n 2 2 t and F 2 tn 2 such that ρ ( a ) = ρ (( a 1 , · · · , a n )) = ( a 1 ( e ) , · · · , a n ( e ) ) , where a i ( e ) (1 ≤ i ≤ n ) is a vector of length 2 t over F 2 and T r F 2 2 t / F 2 ( a · b ) = T r F 2 2 t / F 2 ( P n i =1 a i b i ) = P n i =1 ( a i ( e ) , b i ( e ) ) , wh ere ( · ) stands for the o rdinary scalar produ ct over F 2 2 t . Th us we have Lemma 6 : Let C b e a self-orth ogona l co de over F 2 2 t , then ρ ( C ) is a self-orthog onal code over F 2 . T ABLE II E X A M P L E 3 t equati ons for R a nd δ R = 1 / 2 2 R + 4 δ = 2 3 δ ≈ 0 . 0417 3 R + 6 δ = 6 7 δ ≈ 0 . 0595 4 R + 8 δ = 14 15 δ ≈ 0 . 05417 5 R + 10 δ = 30 31 δ ≈ 0 . 04677 Pr oof: F o r any two co dew ords ρ ( a ) and ρ ( b ) o f ρ ( C ) ( ρ ( a ) , ρ ( b )) = n X i =1 ( a ( e ) i , b ( e ) i ) = T r F 2 2 t / F 2 ( a · b ) = 0 the last equality holds because C is a self-o rthogo nal code. T o show our constructio n, we also ne ed th e result of self- orthog onal codes from [11]: Lemma 7 : ([1 1]) Let q = l 2 be a sq uare. Th en the class o f self-ortho gonal codes me et the Tsfasman-Vl ˇ adut ¸ -Zink b ound. More precisely , we have the fo llowing holds. • Let 0 ≤ R ≤ 1 / 2 a nd δ ≥ 0 with R = 1 − δ − 1 / ( l − 1) . Then the re is a sequenc e ( C j ) j ≥ 0 of linear cod es C j over F q with parameters [ n j , k j , d j ] such that the following: 1) all C j are self-ortho gonal codes; 2) n j → ∞ as j → ∞ ; 3) lim j →∞ k j /n j ≥ R and lim j →∞ d j /n j ≥ δ . Remark 2 : The existence of the self-orth ogonal co des in Lemma 7 is constructive. For the detail of the construction of this codes, we ref er to [11]. Then by Lemm a 7 , we know that there exists a class of self-ortho gonal codes over F 2 2 t which meet the Tsfasman- Vl ˇ adut ¸ -Zink bo und. Now , we g i ve the characterizatio n of ou r construction . Pr opo sition 2: L et C i be a family of self-orthogon al c odes over F 2 2 t which meets the Tsfasman-Vl ˇ adut ¸ -Zink boun d with parameters [ n i , k i , d i ] , i.e., lim i →∞  k i n i + d i n i  = 1 − 1 2 t − 1 . Then ρ ( C i ) is a family of self-o rthogo nal codes over F 2 with parameters [2 tn i , 2 tk i , d i ] . Moreover , we have equation R + 2 tδ = 1 − 1 2 t − 1 , (7) where R and δ d enote the inf ormation r ate and relative minimum distance, re spectiv ely , of the co des ρ ( C i ) . Example 3 : Using this constructio n, we get the equa tions of R and δ in T able II. T he secon d column of T able II was calculated by (7). In particular, it is easy to see that when we c hoose t = 3 and R = 1 / 2 , we get the best value of δ , δ ≈ 0 . 0595 from (7 ) (fo r asymptotic Gilber t-V arshamov bound , we h av e δ ≈ 0 . 1 10 ). I V . G I L B E R T - V A R S H A M OV B O U N D In this section, by mimicking the idea in [ 2], we give the proo f that there exists a family of binary self-orth ogonal codes achieving the Gilbert-V ar shamov bou nd. F or binary self- orthog onal code, it is easy to kn ow that the weight of every codeword is e ven. Now we assume that the length of co de n is also an even number . 4 W e first introdu ce two notation s. L et A be th e set of self- orthog onal cod es o f length n over F 2 , an d let A 1 denote the subset o f A con sisting of all self-dual cod es of len gth n over F 2 . Lemma 8 : ([2]) Let n = 2 h and, le t C be an [ n, s ] b inary self-ortho gonal code. The numb er of cod es in A 1 which contain C is (2 h − s + 1)(2 h − s − 1 + 1) · · · (2 2 + 1)(2 + 1) . Let σ n,k,s , s ≤ k < h , be the number o f self-o rthogon al codes D with p arameters [ n, k ] which contain the given co de C . In the proof of Lemma 8 , the authors establish a recursion formu la for σ n,k,s . σ n,k +1 ,s = σ n,k,s × 2 n − 2 k − 1 2 k − s +1 − 1 . (8) Then we h av e Cor ollary 1: The number of codes in A of d imension k is (2 n − 2( k − 1) − 1)(2 n − 2( k − 2) − 1) · · · (2 n − 1) (2 k − 1)(2 k − 1 − 1) · · · (2 − 1 ) . (9) Pr oof: It is easy to know th at every self-o rthogon al code with dimension k must co ntain the trivial co de 0 . Let s = 0 , th en σ n,k, 0 is th e nu mber we require. Using the re cursion formu la we get (9). Cor ollary 2: Let v be a v ector other than 0 , 1 with wt( v ) ≡ 0( mod 2 ) . The number o f codes in A of d imension k which contain v is (2 n − 2( k − 1) − 1)(2 n − 2( k − 2) − 1 ) · · · (2 n − 2 − 1) (2 k − 1 − 1)(2 k − 2 − 1) · · · (2 − 1) . (10) Pr oof: It is easy to know th at every self-o rthogon al code containing the vector v m ust con tain the cod e C , where C is the linear code with basis { v } . T hen σ n,k, 1 is the numb er we require. Using the recu rsion for mula we get (10). Using th ese two Corollarie s, we hav e Theor em 1: Let r be a p ositiv e integer su ch that  n 2  +  n 4  +  n 6  + · · · +  n 2( r − 1)  < 2 n − 1 2 k − 1 . (11 ) Then the re exists an [ n, k ] self-orthog onal code with minimu m distance at least 2 r. Pr oof: T he theorem is a n immediate conseq uence o f Corollaries 1 a nd 2 . Remark 3 : For a ny 0 ≤ δ ≤ 1 / 2 , let r = ⌊ δn 2 ⌋ , then k = $ log 2 2 n − 1  n 2  +  n 4  + · · · +  n 2( r − 1)  !% satisfy (11), i.e., there exists an [ n, k , 2 r ] binary self- orthog onal code and asy mptotically , we have k n → 1 − H 2 ( δ ) . (12) By Lemma 5 , (12) imp lies that the bin ary self-ortho gonal code meets the Gilber t-V arshamov bound. 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 relative minimum distance imformation rate our constructive bound Gilbert−Varshamov bound Fig. 1. Asymptotic bound on s elf-ort hogonal c odes V . C O N C L U S I O N Using th ese two construction s, we get a sequence o f equa- tions on R and δ . Then we get a constructive b ound o n α 2 ( δ ) by com bining th e equation s (5) and (7). W e draw the figu re of this boun d in Fig .1. When R → 0 , th e constructive boun d (5) is better than th e constructive b ound (7). When R → 1 / 2 , the constructive b ound (7) is better than th e constructive b ound( 5). In Section IV , we proof tha t b inary self- orthogo nal codes meet the Gilbert-V arshamov bound, we also show the figure o f this bound fo r self-o rthogo nal codes in Fig.1. A C K N O W L E D G M E N T The auth or is gratefu l to Profs. Keqin Feng, Jianlo ng Chen and Chaoping Xin g fo r their guida nce. R E F E R E N C E S [1] M. A. Tsfasman and S . G. Vl ˘ a dut ¸, Alge braic- Geometriec Codes. Dor- drecht, The Netherla nds: Kluwer , 1991. [2] F . J. Macwilli ams, N. J. Sloane and J . G. 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