Error-Correction Capability of Reed-Muller codes
aa r X i v : . [ c s . I T ] F e b Error-Correction Capability of Reed-Mullercodes
St´ephanie Dib, Fran¸cois Rodier ∗ Abstract
We present an asymptotic limit between correctable and uncor-rectable errors on the Reed-Muller codes of any order. This limit istheoretical and does not depend of any decoding algorithm.
Let F be the field with 2 elements, and let RM = RM ( n, r ) be the Reed-Muller code of length 2 n and of order r that is the set of Boolean functionwith n variables of algebraic degree not more than r .Building a code is important, but we must think about how many wordswe can decode. Usually, we content ourselves of the fact that errors of weightsless than half of the minimum distance can be corrected in a unique manner.So decoding an error correcting code beyond half of the minimum distancehas been a challenge for the one who study error correcting codes. In factexperiments show that a maximum likelihood decoding can decode manymore words.Here we propose a theoretical bound for decoding almost all errors ofReed-Muller code on any order by this method of decoding. Indeed, thedecoder will often be able to recover the correct codeword using an algorithmthat generates for each received word the closest codeword even if the receivedword is more distant than half of the minimum distance. On the contrary,when the number of errors exceeds a certain value, the received vector willbe rarely closer to the correct codeword than to any other one. Here we givea proof for that. ∗ Aix Marseille Universit´e, CNRS, Centrale Marseille, Institut de Math´ematiquesde Marseille, UMR 7373, 13288 Marseille, France, [email protected],[email protected]
1t is interesting to compare that fact with the phenomenon of concen-tration of the nonlinearity of Boolean functions which have been studied byseveral authors ([3, 4, 9, 11, 12, 13]. The r -nonlinearity of a Boolean function f denoted N L r ( f ) is its Hamming distance to the set of Boolean functionswith n variables of algebraic degree not more than r . Claude Carlet [2],proved that the density of the set of Boolean functions satisfying N L r ( f ) > n − − c s n − (cid:18) nr (cid:19) log 2tends to 1 when n tends to infinity, if c >
1. The authors of the present paperproved a concentration of the nonlinearities of almost all Boolean functionsaround 2 n − − s n − (cid:18) nr (cid:19) log 2 (1)when r ≤ r thanks to a result of Kaufman, Lovett, and Porat [8] helping him to find abetter bound for the weights of a RM code [13].On the other hand, Helleseth, Klove and Levenshtein in the paper Errorcorrection capability of binary linear codes [7] study order 1 or 2 Reed-Mullercodes and they show that almost all the words are decodable up to the samebound as (1) and almost all words are not decodable beyond this bound.For that, they use the monotone structure of correctable and uncorrectableerrors. St´ephanie Dib [4, Chapter 3] proved by the same method as for theconcentration of the nonlinearities of almost all Boolean functions that thebound for correcting most of the values of codewords for 1-order RM codeswas given by (1).We show here that the value given in (1) is also the bound for correctingmost of the values of codewords for RM codes for any order. For RM codes,the present work improves the paper by Helleseth et al. [7] where they justprove the fact that the codes RM ( n, r ) are asymptotically optimal for r = 1(cf. note after inequality (54) of [7]) or r = 2 (example 7 of [7]). Let d ( e, f ) be the Hamming distance between the elements e and f in F n .We denote by wt ( e ) the weight of an element e in F n . Let C be a linearcode of length m , of dimension k . The Reed-Muller code of length 2 n and oforder r has dimension P r (cid:16) nr (cid:17) and minimum distance 2 n − r .2 .1 Correctable and uncorrectable errors Let F n be the set of all binary vectors of length 2 n . For any vector f ∈ F n ,the set f + C = { f + g | g ∈ C } is called a coset of C and contains 2 k vectors. One can easily check that twocosets are either disjoint or coincide. This means f ∈ h + C = ⇒ f + C = h + C. Therefore, the set F n can be partitioned into 2 n − k cosets of C : F n = n − k − [ i =0 ( f i + C ) , f i ∈ F n where ( f i + C ) ∩ ( f j + C ) = ∅ for i = j .If you send a word g and the decoder receive the word h , we will call e = g − h the error. Thus, the possible error vectors are the vectors in the cosetcontaining h . In maximum-likelihood decoding, the decoder’s strategy is,given h , to choose a minimum weight vector e in h + C , and to decode h as h − e .The minimum weight vector in a coset is called the coset leader, and whenthere is more than one vector of minimum weight in a coset, any one of themcan be selected as the coset leader.We denote the set of all coset leaders by E ( C ) (note that E ( C ) = 2 n − k ).The elements of E ( C ) are called correctable errors, and the elements of E ( C ) = F n − E ( C ) are called uncorrectable errors. Only coset leaders arecorrectable errors, which means that 2 n − k errors can be corrected with thisdecoding.A codeword is an unambiguous correctable error if it is a coset leader, andit is the only vector of minimum weight in this coset. Proposition 1 .The following statements are equivalent. A codeword e is an unambiguous correctable error; ∀ e ′ ∈ e + C if e = e ′ then wt ( e ) < wt ( e ′ ) ; ∀ g ∈ C − { } , wt ( e ) < wt ( g + e ) ; ∀ g ∈ C − { } , d ( e, < d ( g, e ) . roof The first assertion implies the second because if e ′ ∈ e + C and e = e ′ then e ′ is not the coset leader, so wt ( e ) < wt ( e ′ ).The second assertion implies the first because if e ′ ∈ e + C and e = e ′ then wt ( e ) < wt ( e ′ ) so e ′ is not the coset leader and e is the only vector ofminimum weight in this coset.The other statement are clear. We take F n as the probability space. We endow it with the uniform proba-bility P . Let λ n = c × n/ r (cid:16) nr (cid:17) log 2 and δ = 2 n − − λ n / c is a positive real.We will show that if c > δ arecorrectable, when n tends to infinity. And that if c < δ are uncorrectable, when n tends to infinity. Moreprecisely we will show the following two theorems. Theorem 1 . Let c > . Then P wt ( e ) ≤ δ (cid:16) d ( e, < d ( e, g ) for all g ∈ RM ( r ) − (cid:17) → when n → ∞ . and Theorem 2 . Let c < . Then P wt ( e ) ≥ δ (cid:16) there exists g ∈ RM ( r ) − such that d ( e, ≥ d ( e, g ) (cid:17) → when n → ∞ . We intend to prove that almost all error of weight smaller than δ for c > n tends to infinity. It is enough to prove P wt ( e ) ≤ δ (cid:16) d ( e, < d ( e, g ) for all g ∈ RM ( r ) − (cid:17) → n → ∞ .
4e have just to show P wt ( e ) ≤ δ (cid:16) δ < d ( e, g ) for all g ∈ RM ( r ) − (cid:17) → n → ∞ or P wt ( e ) ≤ δ (cid:16) ∃ g ∈ RM ( r ) − , δ ≥ d ( e, g ) (cid:17) → n → ∞ that is P wt ( e ) ≤ δ [ g ∈ RM ( r ) − (cid:16) δ ≥ d ( e, g ) (cid:17) → n → ∞ . It is enough to prove that X g ∈ RM ( r ) − P wt ( e ) ≤ δ ( δ ≥ d ( e, g )) → n → ∞ . By expressing the conditional probabilities we have to show that X g ∈ RM ( r ) − P (cid:18)(cid:16) d ( e, ≤ δ (cid:17) ∩ (cid:16) d ( e, g ) ≤ δ (cid:17)(cid:19) P (cid:16) d ( e, ≤ δ (cid:17) → n → ∞ . Let B δ ( g ) be the ball of center g and of radius δ that is the set of e such that d ( e, g ) ≤ δ . The event B δ ( g ) is the set of words f in F n such f ∈ B δ ( g ),that is d ( f, g ) ≤ δ .Hence Theorem 1 is a consequence of the following proposition. Proposition 2 . If c > then X g ∈ RM ( r ) − P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) P ( B δ (0)) → when n → ∞ Before the proof of this Proposition we have to evaluate the terms in thesum.
Lemma 1 . For every real s , one has P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) ≤ exp (cid:16) s (2 n − wt ( g )) − sλ (cid:17) . Proof .Replace δ by its value. P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) = P (cid:16) ( wt ( f ) ≤ δ ) ∩ ( wt ( f + g ) ≤ δ ) (cid:17) = P (cid:16) (2 n − − wt ( f ) ≥ λ/ ∩ (2 n − − wt ( f + g ) ≥ λ/ (cid:17) n − wt ( f ) = X x ∈ F n ( − f ( x ) , n − wt ( f + g ) = X x ∈ F n ( − f ( x )+ g ( x ) . Hence this gives using Markov’s inequality: P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) = P X x ∈ F n ( − f ( x ) ≥ λ ∩ X x ∈ F n ( − f ( x )+ g ( x ) ≥ λ = P exp s X x ∈ F n ( − f ( x ) ≥ exp( sλ ) ∩ exp s X x ∈ F n ( − f ( x )+ g ( x ) ≥ exp( sλ ) ≤ E exp (cid:16) s X x ∈ F n ( − f ( x ) (cid:17) exp (cid:16) s X x ∈ F n ( − f ( x )+ g ( x ) (cid:17) , exp( sλ ) Since the random values f ( x ) are independant P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) ≤ E exp (cid:16) X x ∈ F n s ( − f ( x ) (cid:16) − g ( x ) (cid:17) , exp( sλ ) ≤ Y x ∈ F n E (cid:16) exp (cid:16) s ( − f ( x ) (cid:16) − g ( x ) (cid:17)(cid:17) , exp( sλ ) Because the random values f ( x ) takes the values ± / P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) ≤ Y x ∈ F n cosh (cid:16) s (cid:16) − g ( x ) (cid:17)(cid:17) , exp( sλ ) As cosh( t ) ≤ exp( t / P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) ≤ Y x ∈ F n exp (cid:18) s (cid:16) − g ( x ) (cid:17) / (cid:19) , exp( sλ ) ≤ exp s (cid:16) n + X F n ( − g ( x ) (cid:17) , exp( sλ ) ≤ exp (cid:16) s (2 n − wt ( g )) − sλ (cid:17) . .1 Case where the distances are close to n − . We give a bound for P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) when the distance to 0 of the center g is rather close to 2 n − . Lemma 2 . If | n − − d ( g, | ≤ n − / (cid:18) nr (cid:19) then: P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) ≤ c (( nr ) − ) . Proof .From lemma 1 we have P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) ≤ exp (cid:18) s (cid:16) n + 2 n − wt ( g ) (cid:17)(cid:19), exp( sλ ) ≤ exp (cid:18) s n (cid:16) . (cid:16) nr (cid:17) (cid:17)(cid:19), exp( sλ ) We take s = λ/ n . P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) ≤ exp (cid:18) λ − n (cid:16) .(cid:16) nr (cid:17)(cid:17) (cid:19)(cid:30) exp( λ / n − ) ≤ exp (cid:18) c (cid:16) nr (cid:17) log 2 (cid:16) .(cid:16) nr (cid:17)(cid:17) (cid:19)(cid:30) exp (cid:16) c (cid:16) nr (cid:17) log 2 (cid:17) Simplifying the two members of this fraction by exp (cid:16) c (cid:16) nr (cid:17) log 2 (cid:17) you get P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) ≤ exp (2 c log 2)exp(2 c (cid:16) nr (cid:17) log 2) ≤ c c × ( nr ) n − . We use the follwing lemma, which is an application of a result by Kaufman,Lovett, and Porat [8].
Lemma 3 . Let α be a strictly positive real number. The number B r,n offunctions g in RM ( r, n ) satisfying | wt ( g ) − n − | ≥ n − / (cid:18) nr (cid:19) fulfills B r,n ≤ α ( nr ) if n is large enough. roof This is shown in the proof of Lemma 3 in K.-U. Schmidt’s article [13, relation(6)].We use this lemma to evaluate Π = P P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) where the sum ison the nonzero g in RM ( n, r ) fulfilling | wt ( g ) − n − | ≥ n − . (cid:18) nr (cid:19) . Lemma 4 . Let α be a strictly positive real number. Then Π < α ( nr )2 − c − − r ( nr ) Proof .From lemma 1, for all s , we have P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) ≤ exp (cid:16) s (2 n − wt ( g )) − sλ (cid:17) Let us take s = λ n +1 − wt ( g ). We have, expressing the value of λ andnoting that wt ( g ) is not less than the minimum distance 2 n − r of RM ( n, r ): P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) ≤ exp − λ n +1 − wt ( g ) ! ≤ exp − c × n +1 (cid:16) nr (cid:17) log 22 n +1 − n − r +1 ≤ − c × ( nr ) − − r . Therefore Π ≤ B r,n − c − − r ( nr ) ≤ α ( nr )2 − c − − r ( nr ) . P ( B δ (0)) Proposition 3 . Let r be a fixed integer, δ = 2 n − − c r n − (cid:16) nr (cid:17) log 2 where c is a positive constant. We have P ( B δ (0)) = 12 π − c ( nr )2 c r(cid:16) nr (cid:17) log 2 (1 + o (1)) (2) when n tends to infinity. Lemma 5 . Let n be a positive integer and k ≤ n . Then k X i =0 (cid:18) ni (cid:19) < n · exp − ( n − k ) n ! . When k is sufficiently close to n , the following lemma (see [6, chapter IX,(9.98)], [5, chapter VII]) gives an asymptotic estimation for (cid:16) nk (cid:17) : Lemma 6 . Let n be a positive integer and | n − k | ≤ n . Then (cid:18) nk (cid:19) = 2 n √ π · n · exp − ( n − k ) n ! · (1 + o (1)) , (3) where the term o (1) is independent of the choice of k . Proof of the Proposition.
The number of Boolean functions whose Hamming distance to 0 is boundedfrom above by some number δ equals X ≤ i ≤ δ (cid:18) n i (cid:19) . Thus we have X ≤ i ≤ δ (cid:18) n i (cid:19) = X ≤ i< n − − ( n −
1) 58 (cid:18) n i (cid:19) + X n − − ( n −
1) 58 ≤ i ≤ δ (cid:18) n i (cid:19) . The first sum on the right hand side is taken care of by lemma 5 which willshow that it is negligible with respect of the second sum.To estimate a lower bound of the second sum (which we denote S ), we use(3) S = 2 n √ π n − · (1 + o (1)) · X n − − ( n −
1) 58 ≤ i ≤ δ exp − (2 n − − i ) n − ! . We use that the function in the sum is monotonous to replace the sum by anintegral. S = 2 n √ π n − · (1 + o (1)) · Z n − − ( n −
1) 58 +1 ≤ i ≤ δ exp − (2 n − − i ) n − ! di. = 2 n √ π · (1 + o (1)) · Z c q ( nr ) log 2 ≤ v ≤ n − − − n exp (cid:16) − v (cid:17) dv.
9y [5, chapter VII, Lemma 2] and the fact that c (cid:18) nr (cid:19) log 2 − (cid:16) n − − − n (cid:17) → −∞ which implies that Z ∞ n − − − n exp (cid:16) − v (cid:17) dv = o Z ∞ c q ( nr ) log 2 exp (cid:16) − v (cid:17) dv the last integral is equivalent toexp (cid:16) − c (cid:16) nr (cid:17) log 2 (cid:17) c r(cid:16) nr (cid:17) log 2 = 2 − c ( nr )2 c r(cid:16) nr (cid:17) log 2 . Thus X ≤ i ≤ δ (cid:18) n i (cid:19) = 2 n √ π − c ( nr )2 c r(cid:16) nr (cid:17) log 2 (1 + o (1)) . Therefore X g ∈ RM ( r ) − P (cid:16) B δ (0) ∩ B δ ( g ) (cid:17) P ( B δ (0)) ≤ O ( n r/ ) (cid:18) nr )2 − c (( nr ) − )2 c ( nr ) + 2 α ( nr )2 − c − − r ( nr )2 c ( nr ) (cid:19) . This tends to 0 because the exponent of 2 is, for the left term (cid:18) nr (cid:19) − c (cid:18)(cid:18) nr (cid:19) − (cid:19) + c (cid:18) nr (cid:19) = − (cid:18) nr (cid:19) c + 2 c → −∞ and for the right term α (cid:18) nr (cid:19) − c − − r (cid:18) nr (cid:19) + c (cid:18) nr (cid:19) = (cid:18) nr (cid:19) α − − r c − − r ! . So just take α < − r c − − r so that this term tends to −∞ . 10 The error correction capability function
Let ǫ C ( t ) the ratio of the number of correctables errors of weight t to thenumber of words of weight t . Let us suppose from now on that the lexico-graphically smallest minimum-weight vectors are chosen as the coset leaders.This involves only the cosets with several minimum weight vectors that isthe ambiguous correctable errors. Then an important property of this ratiois that for any t in the range from half the minimum distance to the coveringradius, ǫ C ( t ) decreases with the growing t as the next lemma says. Lemma 7 .For any [ n, k ] code C and any t = 0 , , . . . , n − ǫ C ( t + 1) ≤ ǫ C ( t ) with strict inequality for t C ≤ t ≤ r C where we set t C = ⌊ ( d C − / ⌋ anddenote the covering radius of C by r C . Proof
See Helleseth et al. [7, Lemma 2]. This property is due to the fact thatthe sets of correctable and uncorrectable errors form a monotone structure,(see, for example, [10, p. 58, Theorem 3.11]) and a result of Bollobas aboutshadows [1, Theorem 3]
For Reed-Muller codes of order r , that is to say RM ( n, r ) we take t = 2 n − − c s n − (cid:18) nr (cid:19) ln 2 . Corollary 1 . If c > , then ǫ C ( t c ) → when n → ∞ . Proof .We know that the ratio of the unambiguous correctable errors (hence also thecorrectable errors) of weight smaller than t c to the words of weight smallerthan t c tends to 1 when n tends to infinity. We have to show that the ratioof the correctable errors of weight exactly t c to the words of weight exactly t c tends to 1 when n tends to infinity.For an RM ( n, r ) code, let2 n √ π − c ( nr )2 c r(cid:16) nr (cid:17) log 2 = A ( c ) . c and suppose that ǫ C ( t c )
1. Then there exists η < ǫ C ( t c ) < η for an infinity of n . If c > c >
1, then among the words ofweights between t c and t c there is only at most a proportion η of correctablewords as the function ǫ C decreases. From Proposition 3 there are about X ≤ i ≤ δ (cid:18) n i (cid:19) = A ( c )(1 + o (1)) . words of weight in [0 , t c ] and X δ ≤ i ≤ δ (cid:18) n i (cid:19) = A ( c )(1 + o (1)) . words of weight between t c et t c . As A ( c ) = o ( A ( c )) there are at most A ( c )(1 + o (1)) + ηA ( c )(1 + o (1)) = ηA ( c )(1 + o (1))correctable words of weights [0 , t c ], which shows that it is impossible thatalmost all words are correctable as says Theorem 1. In the case of RM codes we have a simplification of the proof of the Theorem3 b in [7].
Proposition 4 . If c < , then ǫ C ( t c ) → when n → ∞ . For every t , one has (cf. Lemma 3 of [7]) ǫ C ( t ) × t X i =0 (cid:18) n i (cid:19) ≤ t X i =0 ǫ C ( i ) (cid:18) n i (cid:19) = number of correctable errors of weight smaller than t ≤ total number of correctable errors ≤ n − k . We have, from Proposition 3 t c X i =0 (cid:18) n i (cid:19) = B t c = 2 n − c ( nr )2 c r π (cid:16) nr (cid:17) ln 2 (1 + o (1)) . ǫ C ( t c ) ≤ n − kt c X i =0 (cid:18) n i (cid:19) = 2 c r π (cid:16) nr (cid:17) ln 22 n − c ( nr ) × P ri =0 ( ni ) − n (1 + o (1))= 2 c r π (cid:16) nr (cid:17) ln 22 P ri =0 ( ni ) − c ( nr ) (1 + o (1)) . If c <
1, when n → ∞ then the denominator tends toward infinity, so ǫ C ( t c ) → Remark.