The List-Decoding Size of Reed-Muller Codes

The List-Deco ding Size of Reed -Muller Co des T ali Kaufman ∗ MIT k aufman t@mit.edu Shac har Lo vett † W eizmann Institute of Science shac har .lo v ett@we izmann.ac.il No v em b er 3, 2018 Abstract In this w ork w e study the list-deco ding size of Reed-Muller co des. Given a r eceived word and a distance pa rameter, we ar e interested in b o unding the size of the list of Reed-Muller co dewords that are within t hat distance from the received word. Previous b ounds of Go palan, Kliv ans and Zuck erman [4] on the lis t size of Reed-Muller co des apply only up to the minimum distance o f the co de. In this w ork we provide asymptotic b ounds for the list-deco ding size of Reed-Muller codes that apply for al l distances. Additionally , we study the w eight distribution of Reed-Muller codes. Prior r esults of Kas a mi and T okura [8 ] on the structure of Reed- Muller co dewords up to twice the minim um distance, imply bo unds on the w eight distribution of the co de that a pply only until t wice the minimum distance. W e provide accumulativ e bounds for the weigh t distribution of Reed- Muller co des that apply to a l l distances. 1 In tro duction The p r oblem of list-deco ding an error correcting co de is the f ollo wing: giv en a receiv ed wo rd and a distance p arameter fin d all codewords of the co de that are within the give n distance from the receiv ed word. List-deco ding is a generalizatio n of the more common notion of un ique decoding in whic h th e giv en distance p arameter ens u res that there can b e at most one co dewo rd of the co de that is within the giv en distance from the receiv ed w ord. The notion of list-deco d ing has numerous practical and theoretical implications. The br eakthrou gh results in this field are du e to Goldreic h and Levin [3] and Su dan [10] wh o ga v e efficient list deco ding algorithms f or the Hadamard code and the Reed-Solomon co de. See surveys b y Guruswami [5] and Sud an [11] for further details. In co mplexit y , list-decodable co des are used to p erform hardness amplification of functions [12]. In cryptography , l ist-decod able co d es are u sed to construct h ard-core predicates from one w a y functions [3]. In learning theory , list deco d ing of Hadamard codes implies learning parities with noise [7]. In this pap er w e stud y the question of list-decoding Reed-Muller co des. Sp ecifically , we are in terested in b ounding the list sizes obtained for different d istance parameters for th e list-dec o ding problem. ∗ Researc h su pp orted in part by NS F Awards CCF-0514167 and NS F-0729011. † Researc h supp orted partly by the Israel Science F oundation (grant 1300/ 05). Researc h was conduct ed partly when t he author w as an intern at Microsoft Research. 1 Reed-Muller cod es are ve ry fund amen tal and w ell studied codes. RM ( n, d ) is a lin ear co de, whose co d ew ords f ∈ RM ( n, d ) : F n 2 → F 2 are ev aluations of p olynomials in n v ariables of total degree at most d o ver F 2 . In this wo rk we study the co d e R M ( n, d ) w hen d ≪ n , and are in terested in particular in the case of co nstan t d . The follo wing facts regarding RM ( n, d ) are straigh t-forward: It has blo c k length of 2 n , dimen- sion P i ≤ d  n i  and minim um relativ e d istance 2 n − d 2 n = 2 − d . W e defi n e: Definition 1 (Relativ e w eigh t of a f unction) . Th e relativ e w eigh t of a function/co deword f : F n 2 → F 2 is the fraction of non-zero e lemen ts, wt ( f ) = 1 2 n |{ x ∈ F n 2 : f ( x ) = 1 }| A closely related definition is th e distance b et w een t wo functions Definition 2 (Relativ e distance b et w een tw o f unctions) . The relativ e distance b et ween t wo func- tions f , g : F n 2 → F 2 is defined as dist ( f , g ) = P x ∈ F n 2 [ f ( x ) 6 = g ( x )] The main fo cus of this work is in un derstanding the asymptotic growth of th e list size in list-decod ing of Reed-Muller co des, as a function of the distance parameter. Sp ecifically w e are in terested in obtai ning b ounds on th e follo wing. Definition 3 (List-decoding size) . F or a function f : F n 2 → F 2 let the ball at r elativ e distance α around f b e B ( f , α ) = { p ∈ R M ( n, d ) : dist ( p, f ) ≤ α } The list-deco ding size of R M ( n, d ) at distance α , denoted b y L ( α ), is the maximal size of B ( f , α ) o v er all p ossible functions f , i.e. L ( α ) = max f : F n 2 → F 2 | B ( f , α ) | In a recen t work Gopala n, Kliv ans and Zuck erman [4] pro v e that for distances up to the minimal distance of the code, the list-deco ding size of Reed-Muller cod es remains constan t. Theorem 1 (Theorem 11 in [4]) . L (2 − d − ǫ ) ≤ O  (1 /ǫ ) 8 d  Their result of b oun ding the list-deco din g size of Reed-Muller co des is inherently limited to w ork up to the m in im um d istance of the co d e, since it uses a structur al theorem of K asami and T akura o n Reed-Muller co des [8], whic h implies a boun d on the w eigh t distribution of Reed-Muller co des that w orks up to t wice the minimum d istance of the c o de. Additionally , the w ork of [4] has dev eloped a list-decoding alg orithm for RM ( n, d ) whose r unnin g time is p olynomial in the w orst list-decod ing size and in the b lo c k length of the c o de. Theorem 2 (Th eorem 4 in [4]) . Given a distanc e p ar ameter α and a r e c eive d wor d R : F n 2 → F 2 , ther e is an algorithm that runs in time pol y (2 n , L ( α )) and pr o duc es a list of al l p ∈ RM ( n, d ) such that dist ( p, R ) ≤ α . Since Gopalan et al. could obtain n on-trivial b ou n ds on the list-deco d ing size for distance parameter α that is b oun ded b y the m inim um distance of th e Reed-Muller co de, their algorithm yields m eaningful ru n ning time on ly for α that is less than t wice the minimum distance of the co de. 2 1.1 W eigh t distribution of Reed-Muller c o des A clo se notion to the list-decoding size of Re ed-Muller co de is the w eigh t d istribution of the co d e. Definition 4 (Ac cum ulativ e w eigh t distribution) . T he accum ulativ e w eigh t distrib ution of RM ( n, d ) at a relativ e weig h t α is the n um b er of codewo rds up to this w eigh t, i.e. A ( α ) = |{ p ∈ RM ( n, d ) : wt ( p ) ≤ α }| where 0 ≤ α ≤ 1. It is well- kno wn th at for any p ∈ R M ( n, d ) whic h is not ident ically zero, w t ( p ) ≥ 2 − d . Th us, A (2 − d − ǫ ) = 1 for an y ǫ > 0. Kasami and T okura [8 ] charact erized the co dewords in RM ( n, d ) of weig h t up to t wice the minimal distance of th e co de (i.e up to d istance 2 1 − d ). Based on their c h aracterizatio n one could conclude the follo wing. Corollary 3 (Corollary 10 in [4]) . A (2 1 − d − ǫ ) ≤ (1 /ǫ ) 2( n +1) Corollary 3 and simple lo wer b ounds (wh ic h we sho w later, see Lemma 8 ) sh o w that A ( α ) = 2 Θ( n ) for α ∈ [2 − d , 2 1 − d − ǫ ] for an y ǫ > 0 (and constan t d ). 1.2 Our Results Gopalan et al. [4] left as an open problem th e qu estion of b ounding the list-dec o ding size o f Ree d- Muller co des b eyond the minimal d istance. In p articular, they ask what is the maximal α s.t. L ( α ) = 2 O ( n ) . In this work we answer their question. Sp ecifically we sh o w b ounds on the l ist-decod ing size of Reed-Muller co de for d istances passin g the minimal distance. In fact, w e show that the asym p totic b ehavio r of L ( α ), for all 0 ≤ α ≤ 1. O ur fir st result sho ws that there exist ”cut-off distances”, at whic h the list-decoding size c hanges from 2 Θ( n ℓ ) to 2 Θ( n ℓ +1 ) : Theorem 4 (First main theorem - list-decod ing size) . L et 1 ≤ ℓ ≤ d − 1 b e an inte ger, and let ǫ > 0 . F or any α ∈ [2 ℓ − d − 1 , 2 ℓ − d − ǫ ] L ( α ) = 2 Θ( n ℓ ) and L ( α ) = 2 Θ( n d ) for any α ≥ 1 / 2 . Using Theorem 4 , and Theorem 2 w e obtain the f ollo wing algorithmic result for list-deco ding Reed-Muller cod es from an arbitrary distance. Theorem 5 (List-deco d in g algorithm) . Given a r e c eive d wor d R : F n 2 → F 2 that is at distanc e α fr om R M ( n, d ) , for α ∈ [2 ℓ − d − 1 , 2 ℓ − d − ǫ ] . wher e 1 ≤ ℓ ≤ d − 1 is an inte ger, and ǫ > 0 . Ther e exists an al gorithm that runs in time pol y (2 Θ( n ℓ ) ) and pr o duc e s a list of al l p ∈ RM ( n, d ) such that dist ( p, R ) ≤ α The w eigh t distribution of RM ( n, d ) co des b eyo nd t wice the m inim um distance was widely op en prior to our w ork. See e.g. Researc h Problem (15.1) in [9] and the related discu s sion in that Chapter. In this work we pro vid e asymptotic b ounds for the w eigh t distribution of RM ( n, d ) that app lied for all w eigh ts 2 − d ≤ α ≤ 1 / 2. Sp ecifically , our second main result giv es exact b oun daries on the range of α for w hic h A ( α ) = 2 Θ( n ℓ ) , for an y ℓ = 1 , 2 , ..., d . 3 Theorem 6 (Second m ain T heorem - accum ulativ e weig h t distr ibution) . L et 1 ≤ ℓ ≤ d − 1 b e an inte ger, and let ǫ > 0 . F or any α ∈ [2 ℓ − d − 1 , 2 ℓ − d − ǫ ] A ( α ) = 2 Θ( n ℓ ) and A ( α ) = 2 Θ( n d ) for any α ≥ 1 / 2 . Theorems 4 and 6 are asymptotically tigh t for constan t ǫ > 0. F or sub-constan t ǫ , and α ∈ [2 ℓ − d − 1 , 2 ℓ − d − ǫ ], our b ound giv es: A ( α ) ≤ L ( α ) ≤ 2 O ( n ℓ /ǫ 2 ) W e conjecture this dep end ency on ǫ is not optimal, and the correct dep endency should b e log(1 /ǫ ) instead of 1 /ǫ 2 . W e expand more on that in the bo dy of the pap er. 1.3 T ec hniques The b ound s on the accumulat iv e w eigh t distribution of the Reed-Muller co d e are obtained usin g the follo wing no v el strategy . W e show that a fun ction f : F n 2 → F 2 whose weigh t is b ounded by wt ( f ) ≤ 2 − k (1 − ǫ ) can b e c ompute d as an exp ectation of its k th-deriv ativ es multiplied b y some b ound ed co efficients (Lemma 10). Using standard sampling metho ds we then sh o w (Lemm a 11) that a fu nction f : F n 2 → F 2 whose w eigh t is b ounded b y wt ( f ) ≤ 2 − k (1 − ǫ ) can b e w ell ap p ro ximated b y a constan t num b er c = c ( k , ǫ ) of its k th -deriv ativ es. This imp lies that ev ery RM ( n, d ) co d ew ord of we igh t u p to 2 − k (1 − ǫ ) can b e well appr o x im ated by c = c ( k , ǫ ) of its k th-deriv ativ es. Since the distance b et ween every pair of R M ( n, d ) co d ew ords is at least 2 − d , a go o d enough appr o ximation of a R M ( n, d ) co d ew ord determines the Reed-Muller cod ew ord uniquely . Hence, th e num b er of RM ( n, d ) co dewords up to w eigh t 2 − k (1 − ǫ ), is b ound ed by the n um b er of k th-deriv ativ es to the p o wer of c = c ( k , ǫ ). As RM ( n, d ) co dew ords are p olynomials of degree at most d , their k th-d eriv ativ es are p olynomials of degree at most d − k . There can b e at most Θ (2 n d − k ) such d eriv ativ es. Th u s, the num b er of RM ( n, d ) co dew ords u p to weig h t 2 − k (1 − ǫ ), can b e b ounded b y O (2 n d − k ) c = O (2 c · n d − k ). W e complemen t these upp er b ound estimati ons with matc hing lo wer b oun d s. A similar w ork in this line is the w ork o f Viola and Bo gdano v [2], whic h sho ws that a fu nction f : F n 2 → F 2 whose wei gh t is b ound ed b y w t ( f ) ≤ 1 / 2 − ǫ can b e wel l appro ximated by c = c ( k , ǫ ) of its 1st-deriv ativ es. Note that approximati on b y 1st-deriv ativ es do es not imply in general appro ximation b y k th -deriv ativ es whic h is c rucial for obtaining our b ounds here. The b oun ds on the list-decoding size of Reed-Muller co d es are obtained usin g similar tec hniqu es to the ones used for b ounding the ac cum ulativ e we igh t distributions. 1.4 Generalized Reed-Mu ller Co des The problems of b ounding b oth the accum u lativ e weigh t distribution and the list-deco ding size can b e extended to Ge neralized Ree d-Muller, the co d e of l o w -degree p olynomials o v er large r fields . Ho wev er, our tec hniques f ail to prov e tigh t resu lt in these cases. W e pro vide some partial results for this ca se and mak e a conjecture about the correct b ound s in App endix A. 4 1.5 Organization Although our goal is b ounding the list-decod ing size of Reed-Muller co d es, w e first study the accum u lativ e w eigh t d istr ibution of Reed-Muller co des. The tec hniques we d evelo p are then easily transferred to b ounding also the list- deco ding size. The pap er is organized as follo ws. In Section 2 w e study the weig h t distribu tion of Reed-Muller co des and we pro v e the S econd Main Theorem (Th eorem 6). In Section 3 study the list-deco ding size of Reed-Muller co d es. W e generalize the tec h niques of Section 2 to prov e the First Main Theorem (Theorem 4). In Section A w e study similar questions for Generalized Reed-Muller co de and pro vide non-tigh t b ounds for these c o des. 2 W eigh t distribution of Re ed-Muller cod es In this sec tion we study the w eigh t distribution of Reed-Muller codes, and we pro v e our Second Main Theorem (Theorem 6). Let R M ( n, d ) stand f or the co d e of m ultiv ariate p olynomials p ( x 1 , ..., x n ) o ver F 2 of total d egree at most d . In the f ollo wing n and d will alw a ys stand for the num b er of v ariables and the total degree. W e will assum e that d ≪ n , and study in particular the case of constan t d . Our S econd Main Th eorem (Theorem 6) is a direct corollary of Theorem 7, giving an u pp er b ound on the accumulativ e w eig h t at dista nce 2 ℓ − d − ǫ , and Lemma 8, giving a simp le lo wer b ound at distance 2 ℓ − d − 1 . Theorem 7 (Upp er b ound on the acc um ulativ e weig h t) . F or any inte ger 1 ≤ k ≤ d − 1 , A (2 − k (1 − ǫ )) ≤ c 1 2 c 2 n d − k ǫ 2 wher e c 1 = (1 /ǫ ) O ( d/ǫ 2 ) and c 2 = O ( d/ ( d − k )!) . Imp ortantly, c 1 , c 2 ar e i ndep endent of n , and c 2 is indep endent of ǫ . In p articular for c onstant d we g et that A (2 − k − ǫ ) ≤ 2 O ( n d − k ǫ 2 ) Lemma 8 (Lo w er b ound on the accum ulativ e we igh t) . F or any inte ger 1 ≤ k ≤ d A (2 − k ) ≥ 2 n d − k +1 ( d − k +1)! (1+ o (1)) In th e upp er b ound on A ( α ), while t he dep endence on n is tigh t, we b eliev e the dep endence on ǫ can b e impro v ed. F or k = d − 1 (and constan t d ), t he c haracteriza tion of [8] sho ws that A (2 1 − d − ǫ ) = 2 Θ( n log(1 /ǫ )) W e conjecture that this is the co rrect dep endence on ǫ in all the range: Conjecture 9. L et d b e c onstant. F or any inte ger 1 ≤ k ≤ d − 1 , A (2 − k − ǫ ) = 2 Θ( n d − k log(1 /ǫ )) W e start by pr o ving th e lo wer b oun d. 5 Pr o of of L emma 8. Single out k v ariables x 1 , ..., x k , and let q b e any degree d − k + 1 p olynomials on the remaining n − k v ariables. First, for an y suc h q , the follo wing degree d p olynomial has relativ e w eigh t exactly 2 − k : q ′ ( x 1 , ..., x n ) = x 1 x 2 ...x k − 1 ( x k + q ( x k +1 , ..., x n )) The n um b er of d ifferen t p olynomials q is 2 ( n − k d − k +1 ) = 2 n d − k +1 ( d − k +1)! (1+ o (1)) W e will p ro v e Theorem 7 in the rest of the section. W e start by defining discrete deriv ativ es, whic h will b e our m ain to ol in the proof. Definition 5. Let f : F n 2 → F 2 b y a function. W e d efine the discrete der iv ativ e of f in direction a ∈ F n 2 to b e f a ( x ) = f ( x + a ) + f ( x ) W e define the iterated discrete deriv ativ e of f in directions a 1 , ..., a k ∈ F n 2 to b e f a 1 ,...,a k ( x ) = ( ... (( f a 1 ) a 2 ) ... ) a k ( x ) = X S ⊆ [ k ] f ( x + X i ∈ S a i ) W e note that usually deriv ativ es are defined as f a ( x ) = f ( x + a ) − f ( x ), but since w e are working o ver F 2 , w e can ignore the signs. W e define another notion whic h is central to our proof, namely the bias of a function. Definition 6. T he bias of a function f : F n 2 → F 2 is bias ( f ) = E x ∈ F n 2 [( − 1) f ( x ) ] = P [ f = 0] − P [ f = 1] = 1 − 2 wt ( f ) The follo wing lemma will b e the h eart of our pro of. It shows th at if a function f has weigh t less than 2 − k , then it c an b e computed b y a it s iterated k -d eriv ativ es. Lemma 10 (Main technical lemma) . L et f : F n 2 → F 2 b e a function s.t. wt ( f ) < 2 − k (1 − ǫ ) . Then the func tion ( − 1) f ( x ) : F n 2 → {− 1 , 1 } c an b e written as ( − 1) f ( x ) = E a 1 ,...,a k ∈ F n 2 [ α a 1 ,...,a k ( − 1) f a 1 ,...,a k ( x ) ] wher e α a 1 ,...,a k ar e r e al numb ers, of absolute value of at most 10 ǫ W e will first pr o v e Theorem 7 given Lemma 10, and then turn to pro ve Lemma 10. W e will also need the follo wing w ell-kno wn tec hn ical lemma, which sh o ws ho w to transform calculatio n by a veraging man y functions, to appro x im ation by av eraging few functions. Lemma 11 (Approximati on by sampling) . L et f : F n 2 → F 2 b e a function, H = { h 1 , ..., h t } a set of fu nctions fr om F n 2 to F 2 , s.t. ther e exist c onstants c h 1 , ..., c h t of absolute value at most C , s.t. ( − 1) f ( x ) = E i ∈ [ t ] [ c h i ( − 1) h i ( x ) ] ( ∀ x ∈ F n 2 ) 6 Then f c an b e appr oximate d b y a smal l numb er of the functions h 1 , ..., h t . F or any δ > 0 , ther e exist functions h 1 , ..., h ℓ ∈ H for ℓ = O ( C 2 log 1 /δ ) , and a fu nction F : F ℓ 2 → F 2 , s.t. the r elative distanc e b etwe en f ( x ) and F ( h 1 ( x ) , ..., h ℓ ( x )) is at most δ , i.e. P x ∈ F n 2 [ f ( x ) 6 = F ( h 1 ( x ) , ..., h ℓ ( x ))] ≤ δ The fu nction F is a weig hte d majority, i. e. it i s of the form: F ( h 1 ( x ) , ..., h ℓ ( x )) = sig n ( P ℓ i =1 s i ( − 1) h i ( x ) ℓ ) wher e s ig n ( x ) is define d by sig n ( x ) = 1 if x ≥ 0 and sig n ( x ) = − 1 if x < 0 . Mor e over, we c an have s 1 , ..., s ℓ to b e i nte gers of absolute v alue at most C + 1 . Using Lemmas 10 a nd 11 w e no w prov e Th eorem 7. Pr o of of The or em 7. Fix 1 ≤ k ≤ d − 1. W e will b ound the num b er of p olynomials p ∈ RM ( n, d ) s.t. w t ( p ) ≤ 2 − k (1 − ǫ ). Let p b e any such p olynomial. W e app ly Lemma 10 to p . W e can write ( − 1) p ( x ) as ( − 1) p ( x ) = E a 1 ,...,a k ∈ F n 2 [ α a 1 ,...,a k ( − 1) p a 1 ,...,a k ( x ) ] suc h that | α a 1 ,...,a k | ≤ 10 ǫ . W e now apply Lemma 11 to the set of p olynomials { p a 1 ,...,a k ( x ) : a 1 , ..., a k ∈ F n 2 } with δ = 2 − ( d +2) . W e get th at th ere are ℓ = O ( d ǫ 2 ) d eriv ativ es { p a i 1 ,...,a i k : i ∈ [ ℓ ] } s.t. the distance b et ween p ( x ) and F ( x ) is at most δ , wh ere F ( x ) = sig n ( P ℓ i =1 s i ( − 1) p a i 1 ,...,a i k ( x ) ℓ ) and s 1 , ..., s ℓ are in tege rs of absolute v alue at most O ( 1 ǫ ). W e now m ak e an imp ortant yet simp le observ ation, that will let u s b ound the num b er of lo w w eigh t p olynomials by b ounding the num b er of fu n ctions F ( x ). Giv en any F ( x ) , there can b e at most one p ∈ RM ( n, d ) s.t. dist ( F , p ) ≤ δ . Assu me otherwise that there are tw o p olynomials p ′ , p ′′ ∈ R M ( n, d ) s.t. dist ( p ′ , F ) ≤ δ and dist ( p ′′ , F ) ≤ δ . By the triangle inequalit y dist ( p ′ , p ′′ ) ≤ 2 δ < 2 − d , but this cannot h old if p ′ , p ′′ are t w o differen t p olynomials, since the minimum relativ e distance of RM ( n, d ) is 2 − d . So, if w e b oun d the num b er of different fun ctions F ( x ) of the ab o ve form, w e will also b ound the num b er of p olynomials p of r elativ e weigh t at m ost 2 − k (1 − ǫ ). Consider the terms app earing in F : • W e need ℓ = O ( d ǫ 2 ) deriv ativ es and coefficients to describ e F completely . • Any deriv ativ e p a i 1 ,...,a i k ( x ) is a a p olynomial of degree at most d − k , and so h as at most 2 ( n ≤ d − k ) p ossibilities. • Any coefficien t s i has O ( 1 ǫ ) p ossibilities. 7 Th us, the tot al the n um b er of differen t F ’s is a t most  2 ( n ≤ d − k ) · (1 /ǫ )  O ( d ǫ 2 ) ≤ c 1 2 c 2 n d − k ǫ 2 where c 1 = (1 /ǫ ) O ( d/ǫ 2 ) and c 2 = O ( d/ ( d − k )!). W e n o w turn to prov e the Lemmas required for the pro of of T heorem 7. W e prov e Lemma 10 in Subsection 2.1 and L emm a 11 in Subsection 2 .2. 2.1 Pro of of the main t ec hnical lemma: Lemma 10 Before pr o v in g Lemma 10, we need some claims regarding deriv ativ es. The first claim sho ws that if a function has n on-zero bias, it can b e computed b y an a v erage of its deriv ativ es. Claim 12. L et g : F n 2 → F 2 b e a function s.t. bias ( g ) 6 = 0 . Then: ( − 1) g ( x ) = 1 bias ( g ) E a ∈ F n 2 [( − 1) g a ( x ) ] wher e the i dentity holds for any x ∈ F n 2 . Pr o of. Fix x . W e ha v e: ( − 1) g ( x ) E a ∈ F n 2 [( − 1) g a ( x ) ] = E a ∈ F n 2 [( − 1) g ( x ) − g a ( x ) ] = E a ∈ F n 2 [( − 1) g ( x + a ) ] = bias ( g ) The follo wing claim sh o ws that if a fu nction has lo w w eight , then deriv ativ es of it will also h a v e lo w w eigh t, and thus large bias. Claim 13. L et f : F n 2 → F 2 b e a function s.t. w t ( f ) < 2 − k (1 − ǫ ) . L et a 1 , ..., a s ∈ F n 2 for 1 ≤ s ≤ k − 1 b e any derivatives, and c onsider bias ( f a 1 ,...,a s ) . Then bias ( f a 1 ,...,a s ) ≥ 1 − 2 s +1 − k (1 − ǫ ) . In p articular: 1. If s < k − 1 then bias ( f a 1 ,...,a s ) ≥ 1 − 2 s +1 − k 2. If s = k − 1 then bias ( f a 1 ,...,a s ) ≥ ǫ Pr o of. Consider f a 1 ,...,a s f a 1 ,...,a s = X I ⊆ [ s ] f ( x + X i ∈ I a i ) F or random x , the pr obabilit y that f ( x + P i ∈ I a i ) = 1 is w t ( f ), which is at most 2 − k (1 − ǫ ). Th us b y u nion b ound, P x ∈ F n 2 [ ∃ I ⊆ [ s ] , f ( x + X i ∈ I a i ) = 1] ≤ 2 s − k (1 − ǫ ) In particular it implies that wt ( f a 1 ,...,a s ) = P x ∈ F n 2 [ f a 1 ,...,a s ( x ) = 1] ≤ 2 s − k (1 − ǫ ) and w e get th e b ound since bias ( f a 1 ,...,a s ) = 1 − 2 w t ( f a 1 ,...,a s ). 8 W e no w can pro v e Lemma 10 using Claims 12 a nd 13. Pr o of of L emma 10. L et f : F n 2 → F 2 b e a function s.t. w t ( f ) ≤ 2 − k (1 − ǫ ). Thus bias ( f ) = 1 − 2 wt ( f ) > 0 and b y Claim 12 we can write: ( − 1) f ( x ) = 1 bias ( f ) E a 1 ∈ F n 2 [( − 1) f a 1 ( x ) ] If k = 1 we a re done. Otherw ise by Claim 13, f a 1 also has positive bias, bias ( f a 1 ) ≥ 1 − 2 s +1 − k (1 − ǫ ) > 0 and so ag ain by Claim 12 we can write ( − 1) f a 1 ( x ) = 1 bias ( f a 1 ) E a 2 ∈ F n 2 [( − 1) f a 1 ,a 2 ( x ) ] Th us w e hav e: ( − 1) f ( x ) = 1 bias ( f ) E a 1 ∈ F n 2 [ 1 bias ( f a 1 ) E a 2 ∈ F n 2 [( − 1) f a 1 ,a 2 ( x ) ]] W e can con tin u e this pro cess as long a s w e ca n guaran tee that f a 1 ,...,a s has non-zero bias for all a 1 , ..., a s ∈ F n 2 . By Claim 13 w e kno w this happ ens f or s ≤ k − 1, and th u s w e ha ve: ( − 1) f ( x ) = E a 1 ,...,a k ∈ F n 2 [ α a 1 ,...,a k ( − 1) f a 1 ,...,a k ( x ) ] where α a 1 ,...,a k = 1 bias ( f ) 1 bias ( f a 1 ) 1 bias ( f a 1 ,a 2 ) ... 1 bias ( f a 1 ,...,a k − 1 ) W e no w b ound α a 1 ,...,a k . By Claim 13 w e ge t that: α a 1 ,...,a k ≤ 1 ǫ k − 2 Y s =1 1 1 − 2 s − k +1 ≤ 1 ǫ Y r ≥ 1 1 1 − 2 − r ≤ 10 ǫ 2.2 Pro of of Approxima tion b y sampling Lemma: Lemma 11 Pr o of of L emma 11. C ho ose h 1 , ..., h ℓ uniformly an d in dep end en tly from H . Fix x ∈ F n 2 , and let Z i b e the random v ariable Z i = c h i ( − 1) h i ( x ) and let S = Z 1 + ... + Z ℓ ℓ . W e will use the fact that if | S − ( − 1) f ( x ) | < 1 then sig n ( S ) = ( − 1) f ( x ) . W e first b ound the probabilit y that | S − ( − 1) f ( x ) | > 1 / 4 By regular Cher n off argumen ts for b ounded ind ep endent v ariables, sin ce E [ S ] = ( − 1) f ( x ) and eac h Z i is of absolute v alue of at m ost C , w e get that P h 1 ,...,h ℓ ∈ H [ | S − ( − 1) f ( x ) | > 1 / 4] ≤ e − ℓ 32 C 2 9 (see for exa mple Th eorem A.1. 16 in [1]). In particular for ℓ = O ( C 2 log 1 /δ ) w e get th at P h 1 ,...,h ℓ ∈ H [ | S − ( − 1) f ( x ) | > 1 / 4] ≤ δ Th us b y av eraging argumen ts, there exists h 1 , ..., h ℓ s.t. P x ∈ F n 2 [ | c h 1 ( − 1) h 1 ( x ) + ... + c h ℓ ( − 1) h ℓ ( x ) ℓ − ( − 1) f ( x ) | ≥ 1 / 4] ≤ δ W e n o w round eac h co efficient to a close rational, without damaging the app ro ximation error. The coefficient of ( − 1) h i ( x ) is α i = c h i ℓ . If w e round c h i to the closest in teger [ c h i ], w e get that the co efficien t of ea c h ( − 1) h i ( x ) is c hanged b y at most 1 2 ℓ , and th us the tota l appro ximation is c hanged b y at most 1 / 2. Hence we h a v e: P x ∈ F n 2 [ | [ c h 1 ]( − 1) h 1 ( x ) + ... + [ c h ℓ ]( − 1) h ℓ ( x ) ℓ ) − ( − 1) f ( x ) | ≥ 3 / 4] ≤ δ Th us w e got that P x ∈ F n 2 [ sig n ( [ c h 1 ]( − 1) h 1 ( x ) + ... + [ c h ℓ ]( − 1) h ℓ ( x ) ℓ ) 6 = ( − 1) f ( x ) ] ≤ δ 3 List-deco d ing size of Re ed-Muller c o d es In this s ection w e turn to the problem of b oun ding the list-deco ding size o f Reed-Muller codes, a nd w e pr o v e the First Main Theorem (Theorem 4). W e will see that the same tec hn iques w e u sed in Section 2 to b ound the wei gh t distribution, can b e app lied with minor v ariants to also b oun d the list-decod ing size. The list-deco ding size of a co d e is at least th e accumulativ e w eigh t distribution, i.e. L ( α ) ≥ A ( α ). Ho w ev er, th e list-deco ding size can sometimes b e m uc h larger than the ac cum ulativ e weig h t distribution. Theorem 4 is a d irect corollary of Th eorem 14, giving an upp er b ou n d on the list-decod ing size at distance 2 ℓ − d − ǫ , and th e same low er b ound w e u sed to b oun d th e accum ulativ e w eigh t distribution, obtained in Lemma 8. Theorem 14 (Upp er b ou n d o n the list-decoding size) . F or any inte ger 1 ≤ k ≤ d − 1 , L (2 − k (1 − ǫ )) ≤ c 1 2 c 2 n d − k ǫ 2 + c 3 n ǫ 2 wher e c 1 = (1 /ǫ ) O ( d/ǫ 2 ) , c 2 = O ( d/ ( d − k )!) and c 3 = O ( dk ) . Imp ortant ly, c 1 , c 2 , c 3 ar e indep endent of n , and c 2 , c 3 ar e i ndep endent of ǫ . In p articular for c onstant d we get that L (2 − k − ǫ ) ≤ 2 O ( n d − k ǫ 2 ) 10 Pr o of of The or em 14. The pro of w ill b e similar to the pr o of of Theorem 7. Fix f : F n 2 → F 2 to b e an y fun ction. W e w ill b ound the n um b er of p olynomials p of degree at most d s.t. dist ( p, f ) ≤ 2 − k (1 − ǫ ). Let p ∈ RM ( n, d ) b e such a p olynomial, i.e. dist ( p, f ) ≤ 2 − k (1 − ǫ ). Let g ( x ) = p ( x ) − f ( x ), then w t ( g ) ≤ 2 − k (1 − ǫ ). As in the proof of Theorem 7, w e use the deriv ativ es of g to appro ximate g . Set δ = 2 − ( d +2) . By Lemma 10 there are ℓ = O ( d ǫ 2 ) d eriv ativ es { g a i 1 ,...,a i k : i ∈ [ ℓ ] } s.t. the distance b et ween g ( x ) and F ( x ) is at most δ , where F ( x ) = sig n ( P ℓ i =1 s i ( − 1) g a i 1 ,...,a i k ( x ) ℓ ) Th us w e hav e that F + f app ro ximates p , since: dist ( p, F + f ) = dist ( p − f , F ) ≤ δ As in the pro of of Theorem 7 , giv en F (and f ) there c an b e a t most a single p ∈ RM ( n, d ) s.t. dist ( p, F + f ) ≤ δ , and so if w e will b ound the num b er of functions F we will b oun d the num b er of cod ew ords close to f . Consider the deriv ativ e g a i 1 ,...,a i k ( x ) used in the expression for F . By linearit y of d eriv ation it can b e decomp osed as g a i 1 ,...,a i k ( x ) = p a i 1 ,...,a i k ( x ) − f a i 1 ,...,a i k ( x ) Eac h p a i 1 ,...,a i k ( x ) is a d egree d − k p olynomial, and so has at most 2 ( n ≤ d − k ) p ossibilities. Eac h f a i 1 ,...,a i k ( x ) = P S ⊆ [ k ] f ( x + P j ∈ S a i j ) can b e describ ed by the v alues of a i 1 , ..., a i k ∈ F n 2 , since we hav e access to f , and so has at most 2 k n p ossibilities. Eac h co efficien t s i has O (1 /ǫ ) p ossibilities. Thus, in total the n u m b er of differen t F ’s is at most  2 ( n ≤ d − k ) + k n · (1 /ǫ )  O ( d ǫ 2 ) ≤ c 1 2 c 2 n d − k ǫ 2 + c 3 n ǫ 2 where c 1 = (1 /ǫ ) O ( d/ǫ 2 ) , c 2 = O ( d/ ( d − k )!) and c 3 = O ( k d ). A cknow le dgement. T he second author would lik e to thank his advisor, Omer Reingold, for on- going advice and en cour agemen t. He w ould also lik e to thank Microsoft Researc h for their sup p ort during his in tern ship. References [1] N. Alon and J. Sp encer, The Pr ob abilistic M etho d , Second edition, publish ed by John Wiley , 2000. [2] A. Bogdano v and E. Viola. Pseudorand om bits f or p olynomials via the Gow ers norm. I n the 48th Annual Symp osium on F oundations of Computer Scienc e (FOCS 2007) . [3] O. Goldreic h a nd L. Levin, A h ar d c or e pr e dic ate for al l one way functions , In the Pro ceedings of the 21st ACM Symp osium on Theory of Comp uting (STOC), 1989 . [4] P . Gopalan, A. Kliv ans and D. Zuc k erman , List-De c o ding R e e d Mu l ler Co des over Smal l Fields , In the Proceedings of the 40th A CM Symp osium on T heory of Computing (STOC), 20 08. 11 [5] V. Guruswami, List de c o ding of Err or-Corr e cting Co des , v ol 3282 of Lect ure notes in Compu ter Science, Springer 200 4. [6] T. Kaufm an and S. Lo v ett, Worst c ase to Aver age Case R e ductions for Polynomials , T o ap- p ear in the Pro ceedings of the 49t h Annual S ymp osium on F oundations of Computer Science (F O CS), 2008. [7] E. Kushilevitz and Y. Mansour, L e arning De cision T r e es using the F ourier Sp e ctrum , SIAM Journal of Computing, 22 (6), (1993), pp 1331 -1348 . [8] T. Kasami and N. T okura, On the weight structur e of R e e d-Mul ler c o des , In the IEEE T rans- actions on Information Theory 16 (Issu e 6), 197 0. [9] J. MacWilli ams and N. J. A. Sloane, The The ory of Err or Corr e cting Co des , Amsterdam, North-Holland, 1977 . [10] M. Sudan, De c o ding of R e e d-Solomon c o des b eyond th e err or-c orr e ction b ound , Journ al of Complexit y , 13, (199 7), pp . 180 -193. [11] M. Su dan, List de c o ding: Algorithm s and Applic ations , SIGA CT News, 3 1 (200 0), pp 16-27. [12] M. Sud an , L. T revian, S. V adh an Pseu dor andom Gener ators without the X O R L emma , J. Comput. Syst. Sci., 61 ( 2001), pp 236 -266. A Generalized Reed-Muller co des The problems of b ounding b oth the accum u lativ e weigh t distribution and the list-deco ding size can b e extended to Ge neralized Ree d-Muller, the co d e of l o w -degree p olynomials o v er large r fields . Ho wev er, our tec hn iques fail to p ro v e tight r esult in these cases. W e b riefly d escrib e the r easons b elo w, and giv e some partial results. W e start by making some basic definitions. Let q b e a pr ime, and let GR M q ( n, d ) denote the co de of m u ltiv ariate p olynomials p ( x 1 , ..., x n ) o ver the field F q , of tot al degree at most d . Definition 7. T he relativ e weigh t of a fu n ction f : F n q → F q is the fraction of non-zero e lemen ts, wt ( f ) = 1 q n |{ x ∈ F n q : f ( x ) 6 = 0 }| Definition 8. T he relativ e d istance b et w een tw o f unctions f , g : F n q → F q is defined as dist ( f , g ) = P x ∈ F n q [ f ( x ) 6 = g ( x )] The accum ulativ e w eig h t d istribution and the list-deco ding size are defin ed analogously for GRM q ( n, d ), using the a ppropriate definitions for rela tiv e weig h t and relativ e d istance. W e denote them b y A q and L q . F or eac h 1 ≤ k ≤ d , w e define a distance r k : 1. F or k = 1, let d = ( q − 1) a + b , wh ere 1 ≤ b ≤ q − 1. Define r 1 = q − a (1 − b/q ). 2. F or 2 ≤ k ≤ d − 1, le t d − k = ( q − 1) a + b , w here 1 ≤ b ≤ q − 1. Define r k = q − a (1 − b/q )(1 − 1 /q ). 3. F or k = d , define r d = 1 − 1 /q . 12 W e conjecture that b oth for the accum ulativ e w eight distribu tion and the list-decod ing size, th e distances r k are the thresholds for the e xp onential d ep end en cy in n : Conjecture 15. L et ǫ > 0 b e c onstant , and c onsider GRM q ( n, d ) for c onstant d . Then: • F or α ≤ r 1 − ǫ b oth A q ( α ) and L q ( α ) ar e c onstant s. • F or r k ≤ α ≤ r k +1 − ǫ b oth A q ( α ) and L q ( α ) ar e 2 Θ( n k ) . • F or α ≥ r d b oth A q ( α ) and L q ( α ) ar e 2 Θ( n d ) . Pro ving lo wer b ound s for A q ( r k ) is similar to t he case of RM ( n, d ). Lemma 16 (Lo w er b oun d f or A q ) . F or any inte ger 1 ≤ k ≤ d , A q ( r k ) ≥ 2 Ω( n k ) The problem is pro ving matc hing up p er b ounds. Using d irectly the deriv ativ es metho d w e used to g iv e upp er b ou n ds for RM ( n, d ) giv es the same b ou n ds for GRM q ( n, d ), alas they are not ti gh t for q > 2: A q (2 − k − ǫ ) ≤ 2 O ( n d − k ) If w e w ould lik e to get upp er b ounds closer to the low er b ounds, a natur al approac h w ould b e to generalize Lemma 10 to taking sev eral deriv ativ es in the same dir ection (whic h is p ossible o v er larger fields). This w ould giv e us tight results for some v alues of k , if w e could also generalize Claim 12 to the ca se of taking a m ultiple deriv ativ e in t he same direction. Ho w ev er, we didn ’t find a w a y of doing so. Instead, w e giv e partial results for Conjecture 15 in the t w o ends of the sca le: when α ≤ r 1 − ǫ , and when r d − 1 ≤ α ≤ r d − ǫ (when α ≥ r d Lemma 16 g iv es L q ( α ) and A q ( α ) are b oth exponential in n d ). First, the minimal distance of GRM q ( n, d ) is kno wn to b e r 1 . Th u s, for any ǫ > 0, A q ( r 1 − ǫ ) = 1. Gopalan, Kliv ans and Zuc k er m an [4] pro ve that L q ( r 1 − ǫ ) is constan t w hen q − 1 divides d : Theorem 17 (Corollary 18 in [4]) . Assume q − 1 divides d . Then: L q ( r 1 − ǫ ) ≤ c ( q , d, ǫ ) Mo vin g to the case of r d − 1 ≤ α ≤ r d − ǫ , w e p r o v e: Lemma 18. L et ǫ > 0 b e c onsta nt. then: A q ( r d − ǫ ) ≤ 2 O ( n d − 1 ) W e no w mo ve on to pro ve Lemmas 16 and 18. W e start with Lemma 1 6: Pr o of of L emma 16. W e start by p ro ving for 2 ≤ k ≤ d − 1. Let d − k = ( q − 1) a + b , where 1 ≤ b ≤ q − 1. S ingle out a + 2 v ariables x 1 , ..., x a +2 , and let g b e an y degree k p olynomial on the remaining v ariables. The f ollo wing p olynomial has degree d and w eigh t exactly q − a (1 − b/q )(1 − 1 /q ): g ′ ( x 1 , ..., x n ) =   a Y i =1 q − 1 Y j =1 ( x i − j )     b Y j =1 ( x a +1 − j )   ( x a +2 + g ( x a +3 , ..., x n )) 13 The n um b er of d istinct p olynomial g is 2 Ω( n d ) . The p ro ofs fo r k = 1 and k = d are s imilar: for k = 1, let d = ( q − 1) a + b . Let l 1 ( x ) , ..., l a +1 ( x ) b e an y indep endent linear functions, and consider g ′ ( x 1 , ..., x n ) =   a Y i =1 q − 1 Y j =1 ( l i ( x ) − j )     b Y j =1 ( l a +1 ( x ) − j )   F or k = d , let g b e an y degree d p olynomial on v ariables x 2 , ..., x n , and consider g ′ ( x 1 , ..., x n ) = x 1 + g ( x 2 , ..., x n ). W e no w con tinue to pro v e Lemma 18. W e first mak e some necessary definitions. Definition 9. T he bias of a polynomial p ( x 1 , ..., x n ) o ver F q is defined to b e bias ( p ) = E x ∈ F n q [ ω p ( x )] where ω = e 2 π i/q is a primitiv e q -th ro ot of unit y . Kaufman and Lo v ett [6] prov e that biased low-degree p olynomials can b e decomp osed in to a function of a c onstan t n um b er of lo wer degree polynomials: Theorem 19 (Th eorem 2 in [6]) . L et p ( x 1 , ..., x n ) b e a de gr e e d p olynomial, s.t. | bias ( p ) | ≥ ǫ . Then p c an b e de c omp ose d as a fu nc tion of a c onstant nu mb er of lower de gr e e p olynomials: p ( x ) = F ( g 1 ( x ) , ..., g c ( x )) wher e deg ( g i ) ≤ d − 1 , and c = c ( q , d, ǫ ) . W e will use Theorem 19 to boun d A ( r d − ǫ ) for an y constant ǫ > 0. Pr o of of L emma 18. W e will show that any p olynomial p ∈ GRM q ( n, d ) s.t. wt ( p ) ≤ 1 − 1 /p − ǫ can b e decomp osed as p ( x ) = F ( g 1 ( x ) , ..., g c ( x )) where deg ( g i ) ≤ d − 1, and c dep ends only on q , d and ǫ . Thus the num b er of such p olynomials is b ound ed by the num b er of p ossibilities to choose c degree d − 1 p olynomials, and a function F : F c q → F q . The num b er of suc h p ossibilities is at most 2 O ( n d − 1 ) . Let p b e s.t. wt ( p ) ≤ 1 − 1 /p − ǫ . W e will show there exists α ∈ F q , α 6 = 0 s.t. bias ( αp ) ≥ ǫ . W e w ill then finish b y using Theorem 19 on the p olynomial αp . Consider the bias of αp for r andom α ∈ F q : E α ∈ F q [ bias ( αp )] = E α ∈ F q ,x ∈ F n q [ ω αp ( x ) ] = 1 − wt ( p ) since for x ’s for which p ( x ) = 0, E α ∈ F q [ ω αp ( x ) ] = 1, and for x s.t. p ( x ) 6 = 0, E α ∈ F q [ ω αp ( x ) ] = 0. W e th us get that: E α ∈ F q \{ 0 } [ bias ( αp )] = 1 − q q − 1 wt ( p ) ≥ q q − 1 ǫ So, there m u st exist α 6 = 0 s.t. bias ( αp ) ≥ ǫ . 14