On List Decoding of Alternant Codes in the Hamming and Lee metrics Ido Tal Ron M. Roth Computer Science Department, Technion, Haifa 32000, Israel. 1
Previous Work Berlekamp, 1968: Negacyclic codes for the Lee metric. Roth and Siegel, 1994: Classical decoding of RS and BCH codes in the Lee metric. Sudan, 1997: List decoding for the Hamming metric. Guruswami and Sudan, 1999: Improved list decoding for the Hamming metric. Koetter and Vardy, 2000: Further improvement of list decoding for the Hamming metric. Koetter and Vardy, 2002: List decoding for a general metric. 2
Our Results • A refined analysis of the algorithm in [KV00] to finite list sizes. • The decoding radius obtained for alternant codes in the Hamming metric is precisely the one guaranteed by an (improved) version of one of the Johnson bounds. • A list decoder for alternant codes in the Lee metric. • Unlike the Hamming metric counterpart, the decoding radius of our list decoder is generally strictly larger than what one gets from the Lee-metric Johnson bound. 3
List Decoding Let F be a finite field, and let d be a metric over F n . Let C be an ( n, M, d ) code over F . • A list- ℓ decoder of decoding radius τ is a function D : F n → 2 C such that – Each received word y ∈ F n is mapped to a set (list) of codewords. – The list is guaranteed to contain all codewords in the sphere of radius τ centered at y , D ( y ) ⊇ { c ∈ C : d ( c , y ) ≤ τ } . – The list is guaranteed to contain no more than ℓ codewords, |D ( y ) | ≤ ℓ . • For a fixed ℓ , the bigger τ is, the better. 4
GRS and Alternant Codes • Fix F = GF( q ) and Φ = GF( q m ). • Denote by Φ k [ x ] the set of all polynomials in the indeterminate x with degree less than k over Φ. • Hereafter, fix C GRS as an [ n, k ] GRS code over Φ with distinct code locators α 1 , α 2 , . . . , α n ∈ Φ, and nonzero multipliers v 1 , v 2 , . . . , v n ∈ Φ, that is C GRS = { c = ( v 1 u ( α 1 ) v 2 u ( α 2 ) . . . v n u ( α n )) : u ( x ) ∈ Φ k [ x ] } . • Fix C alt as the respective alternant code over F , C alt = C GRS ∩ F n . 5
Score of a Codeword • Define [ n ] = { 1 , 2 , . . . , n } . • Let M = ( m γ,j ) γ ∈ F,j ∈ [ n ] be a q × n matrix over the set N of nonnegative integers. The score of a codeword c = ( c j ) n j =1 ∈ C alt with respect to M is defined by n � S M ( c ) = m c j ,j . j =1 • Example: 2 0 1 0 0 1 1 4 1 1 M = , c = (0 , 1 , 2 , 3) , S M ( c ) = 8 . 0 4 1 4 4 4 1 0 1 1 3 0 0 0 0 6
Lemma 1 The next lemma is the basis of the list decoder in [KV00],[KV02]. Lemma 1 [KV00] Let ℓ and β be positive integers and M be a q × n matrix over N . Suppose there exists a nonzero bivariate h,i Q h,i x h z i over Φ that satisfies polynomial Q ( x, z ) = � (i) deg 0 , 1 Q ( x, z ) ≤ ℓ and deg 1 ,k − 1 Q ( x, z ) < β , (ii) for all γ ∈ F , j ∈ [ n ] and 0 ≤ s + t < m γ,j , ( γ/v j ) i − t = 0 . � h �� i Q h,i α h − s � � j h,i s t Then for every c = ( v j u ( α j )) n j =1 ∈ C alt , S M ( c ) ≥ β = ⇒ ( z − u ( x )) | Q ( x, z ) . 7
Design Process of a List Decoder for C alt Fix some metric d : F n × F n → R and ℓ . Find an integer β and a mapping M : F n → N q × n such that for the largest possible integer τ , the following two conditions hold for the matrix M ( y ) that corresponds to any received word y , whenever a codeword c ∈ C alt satisfies d ( c , y ) ≤ τ : (C1) S M ( y ) ( c ) ≥ β . h,i Q h,i x h z i over Φ that (C2) There exists a nonzero Q ( x, z ) = � satisfies (i) deg 0 , 1 Q ( x, z ) ≤ ℓ and deg 1 ,k − 1 Q ( x, z ) < β , (ii) for all γ ∈ F , j ∈ [ n ] and 0 ≤ s + t < m γ,j , ( γ/v j ) i − t = 0 . � h �� i Q h,i α h − s � � j h,i s t 8
The Mapping M H ( y ) • Let r and ¯ r be positive integers such that 0 ≤ ¯ r < r ≤ ℓ . • Define the mapping y = ( y j ) j ∈ [ n ] �→ M H ( y ) = ( m γ,j ) γ ∈ F,j ∈ [ n ] , as r if y j = γ m γ,j = , γ ∈ F , j ∈ [ n ] . r ¯ otherwise • Example: F = GF(5), n = 4, y = (0100), r = 7, ¯ r = 4. 2 4 4 4 4 1 4 7 4 4 M H = . 0 7 4 7 7 4 4 4 4 4 3 4 4 4 4 9
A Decoder for the Hamming Metric Until further notice, assume that d ( · , · ) is the Hamming metric. Proposition 2 For integers 0 ≤ ¯ r < r ≤ ℓ , let θ be the unique real such that R H = k − 1 1 � � � ℓ + 1 − r � ¯ r + 1 � � = 1 − ( r − ¯ r )( ℓ + 1) θ + + ( q − 1) . 2 2 � ℓ + 1 n � 2 Given any positive integer τ < nθ , conditions (C1) and (C2) are satisfied for β = r ( n − τ ) + ¯ rτ and M = M H . 10
Maximizing over r and ¯ r • Instead of maximizing θ = θ ( R H , ℓ, r, ¯ r ) over r and ¯ r , we find it easier to maximize R H = R H ( θ, ℓ, r, ¯ r ) for a given θ (and ℓ ). ℓ +1 ⌈ ℓ +1 1 • For 0 ≤ θ ≤ 1 − q ⌉ , the maximizing values are: r = ℓ +1 − ⌈ ( ℓ +1) θ ⌉ and r = ⌈ ( ℓ +1) θ/ ( q − 1) ⌉ − 1 . ¯ • The decoding radius, τ , obtained in this case is exactly the one implied by a Johnson-type bound for the Hamming metric. • As ℓ → ∞ , the value R H ( θ, ℓ ) = max r, ¯ r R H ( θ, ℓ, r, ¯ r ) converges q − 1 θ 2 obtained in [KV00]. q to the expression 1 − 2 θ + 11
The Lee Metric • Denote by Z q the integers modulo q . • The Lee weight of an element a ∈ Z q , denoted | a | , is defined as the smallest nonnegative integer s such that s · 1 ∈ { a, − a } . • The Lee distance between two elements a, b ∈ Z q is | a − b | . • Example: Z 8 0 7 1 6 2 5 3 4 12
The Lee Metric for F = GF( q ) Let F = GF( q ). • How do we extend the Lee metric to F n ? • Fix a bijection �·� : F → Z q . • Define the Lee distance d L : F n × F n → N between two words ( x i ) i ∈ [ n ] and ( y i ) i ∈ [ n ] (over F ) as n � d L � |� x i � − � y i �| . i =1 13
The Mapping M L ( y ) • Let r and ∆ be positive integers such that 0 < ∆ ≤ r . • Define the mapping y = ( y j ) j ∈ [ n ] �→ M L ( y ) = ( m γ,j ) γ ∈ F,j ∈ [ n ] , as m γ,j = max { 0 , r − | ( � y j � − � γ � ) | ∆ } , γ ∈ F , j ∈ [ n ] . • Example: F = GF(5), �·� = Identity, n = 4, y = (0100), r = 7, ∆ = 4. 2 0 3 0 0 1 3 7 3 3 M L = . 0 7 3 7 7 4 3 0 3 3 3 0 0 0 0 • If d L ( c , y ) = τ then S M ( c ) ≥ rn − τ ∆. 14
R L ( θ, ℓ ) for the Lee Metric Define R L ( θ, ℓ ) = max r, ∆ R L ( θ, ℓ, r, ∆), where R L ( θ, ℓ, r, ∆) = � � ∆(1+2 r − (2Λ+1) � r +1 � Λ+1 1 � � ( ℓ +1)( r − θ ∆) − (2Λ+1)+ ∆)+ T , ( ℓ +1 2 ) 2 2 3 Λ = min {⌊ r/ ∆ ⌋ , ⌊ q/ 2 ⌋} , and � r − Λ∆+1 � if Λ = q/ 2 2 T = . 0 otherwise 15
R L ( θ, ℓ ) for the Lee Metric (Continued) • For any fixed 0 < ∆ ≤ ℓ , the maximum of R L ( θ, ℓ, r, ∆) over r is attained for ( ℓ + ∆ λ 2 ) / (2 λ ) � � if λ = q/ 2 r ∆ = , � ( ℓ + ∆( λ 2 + λ )) / (2 λ +1) � otherwise where ��� � � λ = min ℓ/ ∆ , ⌊ q/ 2 ⌋ . • R L ( θ, ℓ ) is piecewise linear in θ , where the intervals correspond to the integer values of ∆ ∈ { 1 , 2 , . . . , ℓ } . 16
Asymptotic Analysis Proposition 3 Define χ L ( q ) = ⌊ 1 4 q 2 ⌋ /q . For 0 < θ ≤ χ L ( q ) , denote by L the unique integer such that L 2 − 1 ≤ θ < L 2 +2 L 3( L +1) , and 3 L let λ = min { L, ⌊ q/ 2 ⌋} . Then, R L ( θ, ∞ ) = lim ℓ →∞ R L ( θ, ℓ ) = 1+2 λ 2 − 6 λθ +6 θ 2 if λ = q/ 2 2 λ + λ 3 . λ +3 λ 2 +2 λ 3 − 6 λθ − 6 λ 2 θ +3 θ 2 +6 λθ 2 otherwise λ +2 λ 2 +2 λ 3 + λ 4 • The decoding radius obtained in the asymptotic case ( ℓ → ∞ ) is generally strictly larger than the one implied by a Johnson-type bound for the Lee metric. 17
R L ( θ, ℓ ) 1 Johnson, ℓ = 7 Johnson, ℓ = ∞ ℓ = 7 ℓ = ∞ 0 θ 1 χ L (5) Figure 1: Curve θ �→ R L ( θ, ℓ ) and the Johnson bound for q = 5 and ℓ = 7 , ∞ . 18
Comparison to Previous Work τ τ 104 40 85 50 our decoder 35 Roth & Siegel 45 non-algorithmic 30 40 25 35 30 20 our decoder Roth & Siegel 25 15 20 15 10 10 5 5 k k 5 10 15 20 25 5 10 15 20 19
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