WO1999045911A2 - Universal reed-solomon coder-decoder - Google Patents

Abstract

A generalized Reed-Solomon coding is characterized by the codeword block size n, the information block size k and the generator polynomial g(x), which is a function of β and r according to the equation: g(x)=(x+βr)(x+βr+1)(x+βr+2)(x+βr+3)(x+βr+4)...(x+βr+2t-1) where β is an element of the Galois field and r is the initial exponent for β. For decoding, an approach to finding error patterns uses a modified general form that takes advantage of properties of the generating polynomial. With the disclosed universal RS encoding-decoding approach, a user only needs to specify four parameters, namely those values (n, k, β, r), and any RS coding design specification can be met using a single framework.

Description

UNTVERSAL REED-SOLOMON CODER-DECODER

BACKGROUND OF THE INVENTION

Reed-Solomon (RS) coding is a non-binary Bose, Chaudhuri and Hocquenghem (BCH) error correction code. Since each codeword (or a symbol) is represented by a byte (i.e., a fixed length sequence of "0" and " 1 "s) rather than a single bit as in binary BCH coding, and since the decoder can correct an erroneous symbol regardless of how many bits within that symbol are corrupted, RS coding is very powerful against burst errors. A Reed-Solomon codec (outer coding) concatenated with a convolutional codec (inner coding), provides the best of both worlds — block coding and convolutional coding. A soft-decision Viterbi decoder can be easily constructed to take in unquantized "soft" signals and provide a coding gain of about 2 dB over hard-decision decoding. However, a Viterbi decoder tends to generate burst errors. On the other hand, a Reed-Solomon decoder has difficulty performing soft-decision decoding, but can easily correct burst errors that exist at the output of the Viterbi decoder.

A well-known reference regarding error control coding in general and Reed- Solomon coding in particular is "Error Control Coding: Fundamentals and Applications", S. Lin and D. Costello, Jr., Prentice-Hall, 1983 (hereinafter "Lin & Costello"). For each Reed-Solomon coding specification, there is an associated Galois field (GF) specified which depends on the symbol size. For instance, if each symbol is an 8-bit byte, then GF(2⁸=256) is used. This field contains 256 elements which are all the possible 8-bit symbols. Each Galois field has an all 0 element and a primitive element that can generate all the remaining elements in this field. A Reed-Solomon coding scheme is specified as RS(n,k,t), where "k" is the number of information symbols in each input block, "n" is the number of encoded codewords in each output block, and "t" is the number of correctable erroneous symbols such that 2t=n-k. For example, in the Digital Video Broadcasting (DVB) standard, the RS coding is specified as RS(«=204, k=\88, t=8), which is a shortened version of RS(w=255, £=239, t=8). In the INTELSAT standard, the RS coding is specified as RS(219,201, t=9), shortened from RS(255,237,t=9). In a well-known cable modem standard, the code RS(«=255, k=223, t=16) is used. To perform RS encoding and decoding, a generating polynomial is also required. Usually, the generating polynomial of RS(n,k,t) is given by: g(x) = (x+cc) (x+α ²) ... (x+α ^2t)

= g₀ + g, X + g₂ x² + g₃ x³ + ... + g_2t x^2t where α is the primitive element in of the Galois field.

SUMMARY OF THE INVENTION

The present invention relates to a method and apparatus for a universal Reed- Solomon encoder and decoder. "Universal" is used herein to indicate that to realize any particular Reed-Solomon coding scheme, modifications to the general codec framework provided by the invention are minimal. Such a universal approach helps to expedite the design process for various Reed-Solomon coding standards.

In the present invention, a generalized Reed-Solomon coding is characterized by the codeword block size n, the information block size k and the generating polynomial g(x), which is a function of β and r according to the equation: g(x)= (x+β^r) (x+β^r+1) (x+β^r+2)(x+β^r+3) (x+β^r+4) ... (x+β^²¹-¹) where β is an element of the Galois field and r is the initial exponent for β.

For decoding, an approach to finding error patterns of the present invention uses a modified general form that takes advantage of properties of the generating polynomial g(x).

In accordance with the present invention, a Reed-Solomon (RS) decoder is operable to decode a received codeword block represented by a codeword polynomial R(x) that has been encoded using a RS code (n,k,t) and the generating polynomial g(x)= (x+β^r) (x+β^¹) (x+β ²)(x+β^r+3) (x+β^r÷ ) ... (x+β^²¹-¹). For notation, n is a number of encoded codewords in the block, k is a number of information symbols, t is a number of correctable erroneous symbols, β is an element of a Galois field, r is the initial exponent for β, and each β^r+l is a root of g(x). The decoder includes a syndrome computer for computing a series of syndromes from the received codeword polynomial R(x) and an error location polynomial computer for computing an error location polynomial of degree v < t using the computed syndromes. The error location polynomial computer uses an iterative Berelkamp table. The decoder further includes means for determining error symbol locations by evaluating the error location polynomial for each element in the Galois field. The decoder also includes means for determining an error pattern for each error symbol by 1) computing an error evaluator polynomial Z(x) based on the error location polynomial and the computed syndromes and 2) computing the error pattern at each error symbol location according to the expression: e_jy = p-^-¹> - z(β-^jy) / π^v _I._lιl,_v (i+ p'-^J where j_y is a particular error location for which the error pattern is being determined and Z(β^"jy) is the value of the error evaluator polynomial Z(x) evaluated with x = β^"jy. This modified general form for determining the error pattern at each error location is universally applicable for arbitrary β, r specified in the aforementioned generating polynomial g(x).

According to an aspect of the invention, the Reed-Solomon decoder further includes means for correcting errors in each error symbol according to the determined error patterns.

With the universal RS encoding-decoding approach disclosed herein, a user only needs to specify four parameters, namely those values (n,k,β,r), and any RS coding design specification can be met using a single framework. This RS codec framework can therefore be applied to numerous applications, including transmission applications using such standards as the DVB, INTELSAT and emerging cable modem standards. This approach is also very fast due to extensive use of lookup tables and Gallager's modified Berlekamp iterative search. BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features and advantages of the invention will be apparent from the following more particular description of preferred embodiments of the invention, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention.

FIG. 1 is a circuit block diagram of a conventional encoder circuit modified in accordance with the present invention. FIG. 2 is a flow diagram in accordance with a decoder embodiment of the present invention.

FIG. 3 is a flow diagram of operation of a syndrome computer of the decoder embodiment of FIG. 2.

FIG. 4 is a flow diagram of operation of an error location polynomial computer of the decoder embodiment of FIG. 2.

FIG. 5 is a flow diagram of an error pattern determination block of the decoder embodiment of FIG. 2.

DETAILED DESCRIPTION OF THE INVENTION

A general form of the generating polynomial of S(n,k,t) is given by:

Eq.(la) g(x) = (x+β^r) (x+β^r+1) (x+β^r+2)(x+β^r+3) (x+β^⁴) ... (x+β^^21"1) ,

where β is an element of the Galois field and r is the initial exponent for β. Thus, each β^r+1 is a root of the generating polynomial g(x).

Equivalently,

Eq.(lb) g(x) = go + g, x + g₂ x² + g₃ x³ + ... + g_2t x^2t For instance, in the DVB standard specified as RS(204,188,t=8), β=α (the primitive element of GF(256) with = (01000000)=2) and r=0; in the INTELSAT standard specified as RS(219,201,t=9), β=ce⁵³ =(00001010)=40, and r=120.

A Reed-Solomon coding scheme is completely specified by RS(n,k,t) and g(x). This is the reason that a universal RS codec can be very versatile for various RS coding specifications in different standards. Once the codec framework has been built, only a few changes are needed to yield another RS coding structure.

As noted in the background, there is an associated Galois field (GF) specified for each Reed-Solomon coding specification which depends on the symbol size. For instance, if each symbol is an 8-bit byte, then GF(2⁸=256) is used. This field contains 256 elements which are all the possible 8-bit symbols. Each Galois field has an all 0 element and a primitive element that can generate all the remaining elements in this field. There is also a primitive polynomial p(x) that defines the arithmetic operations of the Galois field. For example, a GF(8) with primitive polynomial p(x)=l+x+x³ (i.e., p( ) = 1+ +α³ = 0, or α³ = 1+α where is the primitive element of the Galois field and powers of α are referred to as "roots" of the polynomial) is tabulated as follows (with the least significant bit on the left):

000 α° 100 α¹ 010 ² 001 α³ 110 α⁴ 011 ⁵ 111 α⁶ 101

Using this table, any arithmetic operation between two symbols in an RS code can be easily carried out. For the example of GF(8) above, symbol multiplication is by modulo-7 addition of exponents (since α⁷=l) and symbol addition is by bit-wise EXCLUSIVE-OR (XOR). If the bit patterns of two symbols are known and a multiplication operation is required, the corresponding exponents can be found simply from a look-up table (denoted as V2E "value-to-exponential" table) and then the two exponents can be added. Likewise, if the exponents of two symbols are known and an addition operation is required, the corresponding bits can be found from another look-up table (denoted as E2V "exponent-to-value" table) and then the two bytes can be operated on by a bit-wise XOR.

A partial E2V look-up table for GF(256) is given as follows:

E2V[0] 0

E2V[1] 1

E2V[2] 2

E2V[3] 4

E2V[4] 8

E2V[5] 16

E2V[6] 32

E2V[7] 64

E2V[8] 128

E2V[9] 29

E2vrιoι 58

A partial V2E look-up table for GF(256) is given as follows:

V2E[0] -1

V2E[1] 0

V2E[2] 1

V2E[3] 25

V2E[4] 2

V2E[5] 50

V2E[6] 26

V2E[7] 198

V2E[8] 3

V2E[9] 223

V2E[10] 51 Note that the index of the E2V table is the true exponent plus 1 because "E2V[0]=0" indicates or¹ =0, which is a special case for multiplication operations. For example,

"E2V[4]=8" means α³ =8 "V2E[8]=3" means 8=α³

1. Encoding

An embodiment of an encoder 100 is shown in the block diagram of FIG. 1. The encoder 100 is the conventional encoder circuit for cyclic codes disclosed in the above-referenced Lin & Costello with modification. The encoder 100 is a division circuit that includes multipliers 101 weighted by coefficients g₀ , g ..., g_n._k., , adders 102 and storage devices or registers 110. The multipliers 101 each multiply a field element from the Galois field by a coefficient g_;. The adders 102 each add two elements from the Galois field. The registers 110 each store an element bj from the Galois field. Gate_B 122 connects information digits received on input 104 to bus 120. Switch 126 toggles between the input information digits on input 104 and parity-check digits stored in storage devices 110. The conventional encoder circuit is modified to include Gate_A 124, such that when a shorter generating polynomial is specified, Gate_A 124 is turned off and the remaining coefficients are re-indexed as g₀ , g„ ..., g_n._k.ι . In the conventional encoder, parity check digits are the remainder of x^n_k u(x)/g(x), where u(x) is the information polynomial

Eq.(2) u(x)=u₀ + u, X + ... + u_k_, x k¹ -l

that represents the information digits U_Q, U ,,..., u_k., .

The conventional encoding steps are as follows: Step 1. With Gate_B 122 turned on, the k information digits UQ, u,,... u_k., are shifted into the circuit 100 and simultaneously into the communication channel on line 106 with the switch 126 set on for receiving the information digits on input 104. The information digits are shifted from the front, i.e., beginning with u_k_, and ending with u₀. As soon as the complete digits have entered the circuit, the (n-k) digits in the registers 110 form the remainder and become the parity-check digits.

Step 2. Turn off Gate_B 122. The switch 126 is set on for the parity check digits.

Step 3. Shift the parity-check digits out and send them into the communication channel 106. These (n-k) parity-check digits b₀ , b, , ..., b_n._k_, , together with the k information digits, form the complete codeword.

After RS encoding, there are n codewords with the first k codewords being the information bytes and the remaining n-k bytes being the parity check. Each byte contains / bits when the underlying Galois Field is GF(2')- For example, each byte of the codewords is an 8-bit byte when the underlying field is GF(256); each byte of the codewords is a 6-bit byte when the underlying field is GF(64). A coded block of codewords of RS(«,&) at the output of the encoder then appears as shown in the following table:

InfoByte[k-l] InfoByte[k-2]

InfoByte[l] InfoByte[0] ParityByte[n-k-l] ParityByte[n-k-2]

ParityByte[l] ParityByte[0] 2. Decoding

After the codewords are transmitted through the communication channel, the received codewords may contains errors. That is, some or all bits in a certain codeword may be wrong, so that such codeword is said to be an erroneous symbol. An RS(n,k) code can correct up to t-(n-k)/2 erroneous symbols. Decoding is typically performed in five steps: 1) calculating syndromes, which are denoted as S₀, S,, ... S_2t-ι; 2) building a Berlekamp table to find an error location polynomial, which is denoted by σ(x); 3) finding the locations of the erroneous symbols; 4) finding the error patterns at those error locations; and 5) correcting the errors. A flow diagram for an embodiment of an RS decoder in accordance with the present invention is shown in FIG. 2. There are five blocks in the decoder embodiment including a syndrome computer 200, error location polynomial computer 204, error symbol location determination block 206, error pattern determination block 210 and error correction block 212. The syndrome computer 200 uses the roots β^{1 "1} of the generating polynomial defined in Eq.(la). The error symbol location determination block 206 iteratively uses the elements β¹ to find erroneous symbol locations. The operation of each of these blocks is now described further with reference to FIGs. 2 to 5.

Block 200: Syndrome Computation The received codeword polynomial is given by:

Eq.(3) R(x) = Ro + _x x + R₂ x²... + R_n.₂ x"^"2 + R^ x"^"1

Syndromes are computed by:

Eq.(4) S = R(β^r+1) = Ro + R, β^r+' + R₂ β^2*(r+ ... + R_n-2 p*"^{" )*} + R_n., po-D'O⁺ where i takes the values of 0, 1 , 2, ...2t- 1. In other words, the syndromes are the values of the codeword polynomial Eq.(3) with the variable being the roots of the generating polynomial in Eq.(la), i.e.,

Eq.(5) S- Ro + R, x + R₂ x²... + R_π.₂ x^n"2 + R_n., x"^'1 | _x=p ^rt

For example, in the DVB standard the generating polynomial defines β=α

(the primitive element) and r=0, such that the roots of the generating polynomial are 1, α¹, ², ... α¹⁵. The syndromes are then given by:

S₀=R(α°) = R₀ + R₁+R₂+...+R_n.₂ + R_n., S, = R( α¹ ) = Ro + R, α + R₂ ² + ... + R_n.₂ α^(n"2) + R_n., a⁽ⁿ-^1} S₂ = R( α² ) = Ro + R, α² + R₂ α⁴ + ... + R_n.₂ α^(n"2)*2 + R_n., α^{(n"ι 2}

S₁₅= R( α¹⁵ ) = R_Q + R, α¹⁵ + R₂ α³⁰ + ... + R_n.₂ α^(n"2)*15 + R_n., ^{{n *]5}

In the INTELSAT standard, the generating polynomial defines β=α⁵³ and r=120, such that the roots of generating polynomials are ^53*120, α^53*121, ... ^53*135. The syndromes are then given by:

S₀= R( ⁵³'¹²⁰ ) = Ro + R, α^53*120 + R₂ ^53*120*2 + ... + R„.₂ ^53*120 -²> + R„., α^53*120**"^"1' S, = R( α^53*121 ) = Ro + R, α^53*121 + R₂ ^53*121*2 + ... + R_.₂ α^53*121*'"^"2' + R,,., α^53*121*^¹' S₂ = R( ^53*122 ) = Ro + R, α^53*122 + R₂ α^53*122*2 + ... + R_n.₂ α^53*122*'"-²⁾ + R_n.₁ a⁵'^122*^

S₁₅ = R( α^53*135 ) = o + R, ^53*135 + R₂ ^53*135*2 + ... + R„_.2 ^{53*135 "}» + R_n., _α ^53*i3s ⁿ-ⁱ⁾

A faster way to compute syndromes is to rearrange Eq.(5):

Eq.(6) S_i=((((R_n.₁x + R_n.₂)x + R_n )x + R_n.₄)x+...+R₁)x + R₀ | ^ Thus, the syndrome calculation can be carried out iteratively. This iterative approach used by syndrome computer 200 is shown in the flow diagram of FIG. 3. In block 302, the j^th symbol Rx^ is received. The value of the i^th syndrome (i<j) in block 304 is taken to look-up table V2E to find the corresponding exponent format in block 306. Since x = β^r+1, a modulo addition is performed on the exponent of β^¹ and the exponent format from block 306. This result is passed to look-up table E2V to find the corresponding value format which is then added to the j^th symbol Rx_jji. This result is then used to update the value of the syndrome in block 304.

Block 204: Berlekamp Table for Error Location Polynomial The preferred embodiment uses the Berlekamp table as modified by the method disclosed in "Information Theory and Reliable Communication", R. Gallager, John Wiley and Sons, 1968 (hereinafter "Gallager"). An error location polynomial σ(x) is obtained in an iterative manner. The maximum number of iterations for constructing the Berlekamp table is 2t, where t is the number of correctable errors. For example, in the DVB standard, with code RS(255,239,t=8), the maximum number of iterations is 16. However, in practice, if the received symbol block contains less than 2t errors, fewer iterations are needed. In fact, if the number of errors in the received symbol block is v (v < t ), only t+v iterations are needed in the Berlekamp table to obtain the error location polynomial σ(x). At each step, an updated version of σ(x) is obtained. Thus, at step μ of the iteration, the error location polynomial is:

Eq.(7) σ^(μ) (x) = 1 + σ , ^(μ) x + σ₂ ^(μ) x² + ... + σ_lμ . , ^(μ) x^lμ -¹ + σ_lμ ^(μ) x^lμ

where l_μ is the degree of σ^(μ) (x), σ , ^(μ) is the i^th coefficient of the polynomial σ^(μ) (x) and is represented in terms of the exponent of the primitive element . To determine the updated error location polynomial at step μ+1, a quantity called discrepancy is needed. The μ^lh discrepancy d_u is calculated by: Eq.(8) d_μ = S_μ + σ, ^ S_μ.,+ σ₂ ω S_μ_₂ + ... + σ_lμ ., ω S_μ,_μ+1 + σ_lμ oo S_μ,_μ

where S_μ , S_μ-1 , ..., S_μ-lμ are the syndromes. The error location polynomial at step μ+1 is determined by:

Eq.(9) σ^(μ+1) (x) = σ^(μ) (x) + d_μ d_p ^"' x^(μ"p) σ^(p) (x)

where p is a selected row prior to the μ^th row, d_p and σ^(p) (x) are the discrepancy and error location polynomial of the p^throw, respectively. The values d_p , σ^(p) (x) and μ-p are determined by following the procedures developed by Gallager.

The term "mu" is used to index the rows of the Berlekamp table. For example, when mu=0, σ⁽¹⁾ (x) is actually computed; likewise, when mu=l, σ⁽²⁾ (x) is actually computed; so for the DVB standard where the number of correctable errors t=8, the terms σ⁽¹⁾(x), σ⁽²⁾ (x), ... σ^(I6) (x) need to be found, thus mu=0,I,2, ...,I5. The second term in Eq.(9) is defined as Remedy (x), i.e.,

Eq.(lO) Remedy(x) = d_μ d_p-' x^{(μ' )} σ^(p) (x)

Throughout the process of building the Berlekamp table, consideration is made of three integers: l_{μ ,} μ-p, d_p and four polynomials: Temp(x), d^p)(x), d^μ)(x), Remedy(x). Before the iteration begins, the following are initialized:

μ-p = l, d_p= l = α°, σ^{( )}(x) = l, σ^(μ)(x) = l. The iteration then begins. For mu=0, e.g., σ⁽¹⁾ (x) is to be computed. The discrepancy in Eq.(8) is given as: lfd₀= 0, then

update (μ-p): μ-p ^«= (μ-p) + 1,

and the iteration continues with mu=I. If d₀ ≠O, then

(!„,„/ d_p= do/ d_p= do/1 = d₀= S₀ (recall d_p=l as initialized),

Remedy(x) = (d^/ d_p) x^(μ"p) σ< >(x) = d₀x' = S₀x save σ^(μ)(x) in Temp(x): Temp(x) <= σ^(μ)(x) update σ^(μ)(x): σ^(μ)(x) = σ⁽¹⁾(x) «- σ^(μ)(x) + Remedy(x) = 1 + S₀ x compare 2 i_μ with mu :

mu+l- l_μ (here 2 ^■l_μ=0=mu=0, so l_μ-l), μ-p - 1, d_p *- d_rnu (here d_p = do), σ^(p)(x) - Temp(x) (here σ^{( )}(x)=l). if 2-l_μ> mu, then μ-p «= (μ-p) + 1, and do nothing else. • move on to the next iteration with "mu"=2 and repeat the above.

The iteration continues until all the syndromes have been used. The iteration index "mu" therefore ranges from 0, 1, 2, ... 2t-l (maximum). The final form of the error location polynomial obtained at the end of the iterations can be represented as:

Eq.(l l) σ(x) = l + σ₁ x + σ₂ x² + ... + σ_v.₁ x^v-' + σ_vx^v

where v is the degree of the error location polynomial and v < t. The steps of each iteration for the Berlekamp table carried out in block 204 (FIG. 2) for finding the error location polynomial are shown in the block diagram of FIG. 4. In block 402, the discrepancy is computed using σ(x) and the new syndrome S_f. In block 404, Remedy(x) is computed using the discrepancy result, d_p, and σ^(p)(x). The result for σ^(μ)(x) is assigned to Temp(x) in block 406 and σ^(μ)(x) is updated by adding Remedy(x) in block 408. In block 410, a comparison is made between 2-l_μ and mu. If 2-1 _μ is less than or equal to mu, then the parameters in block 414 are updated and assigned. If 2i_μ is greater than mu, then the expression μ-p is only updated in block 412. A helpful way to further understand the above procedure is to step through an example adapted from Lin & Costello. Consider a triple-error-correcting RS code with symbols from GF(2⁴). The syndromes are then:

S₀ = α¹², S,= 1 = α°, S₂= α¹⁴, S₃= α¹⁰, S₄= 0, S₅ = α¹²

Initialization:

l_μ =0, μ-p = 1, d_p =1 =α°, σ^(p)(x) = 1, σ<^μ>(x) =1.

Iteration begins:

for mu=0: do = S₀ = α¹²

Remedy(x) = (d^/ d_p ) x^(μ"p) σ^(p)(x) = dβX¹ = α¹² x Temp(x) = σ^(μ)(x)=l, σ^(μ)(x) = σ⁽¹⁾(x) «= σ^(μ)(x) + Remedy(x) = 1 + α¹² x since 2 -l_μ-0 ≤ mu=0, then l_μ <= mu+ϊ- l_μ =1, μ-p«=l, d_p^d_mu = do=α¹², σ^(p)(x)^Temp(x)=l. After mu=0, we have:

= 1 μ-p = 1

^ = α¹² σ^(μ)(x)=σ⁽¹⁾(x) = 1 + α^I2x σ^(p)(x) = 1

Remedy(x) = α¹² x

Temp(x) = 1

formu=l: d, = S, + S₀σ,^l)= 1 + α¹²-α¹²=α⁷ Remedy(x) = (^ d_p ) x^{(μ" )} σ^(p)(x) = ( , / d_p ) x¹ = α¹⁰ x

Temp(x) = σ^(μ)(x) = σ⁽¹⁾(x) =1 + α¹² x, σ^(μ)(x) = σ⁽²⁾(x) <= σ^(μ)(x) + Remedy(x) = 1 + α¹² x + α¹⁰ x = 1 + α³x. since 2i_μ=2 > mu=I, then (μ-p) ^«= (μ-p)+l = 2.

After mu=l, we have:

l = 1 (unchanged) μ-p = 2

^ = α¹² (unchanged) σ^(μ)(x)=σ⁽²⁾(x) = 1 + α³ x σ^(p)(x) = 1 (unchanged)

Remedy(x) = α¹⁰x

Temp(x) = 1 +α¹²x

formu=2: d₂ = S₂ + S, σ,⁽²⁾= α¹⁴ + 1- α³= 1 Remedy(x) = (c ,, / d_p ) x^(μ"p) σ^(p)(x) = (d₂ / d_p ) x² = α³ x²

Temp(x) = σ^(μ)(x) = σ⁽²⁾(x) =1 + α³ x, σ^(μ)(x) = σ⁽³⁾(x) <= σ^(μ)(x) + Remedy(x) = 1 + α³ x + α³ x² since 2 -l_μ=2 ≤ mu=2, then l_μ*= mu+l- l_μ =2, μ-p *^*" 1, d_p^d_mu = d₂=l, σ^(p)(x) <= Temp(x) =1 + α³ x.

After mu=2, we have:

l^μ = 2 μ-p = 1

_σ(μ)(_χ)=_σ(3)(_χ) = 1 + α³ x + α³ x² σ*>(x) = 1 + α³ x

Remedy(x) = α³x²

Temp(x) = 1 + α³ x

formu=3: d ^A3, = ° S3, + ' ° S,2σ^υ,l⁽³⁾ + ' S °,1 σ₂ _ ^υ2⁽³⁾ = α¹⁰ + α¹⁴- α³+ 1- α 3 α'

Remedy(x) = (d_^ / d_p ) x^(μ"p) σ^(p)(x) = (d₃ / d_p ) x¹ (1 + α³ x)

= ⁷x + ¹⁰x² Temρ(x) = σ^(μ)(x) = σ⁽³⁾(x) = 1 + α³ x + α³ x² σ^(μ)(x) = σ⁽⁾(x) <= σ^(μ)(x) + Remedy(x) = 1 + α⁴ x + α¹² x² since 2i=4 > mu=3, then (μ-p) *=(μ-p)+l =2. After mu=3, we have:

= 2 (unchanged) μ-p = 2 dp = 1 (unchanged) σ^(μ)(x)=σ⁽⁴⁾(x) = 1 + α⁴x + α¹²x² σ^(p)(x) = 1 + α³ x (unchanged)

Remedy(x) = α⁷ x + α¹⁰ x²

Temp(x) = 1 + α³ x + α³ x²

for mu=4: d₄ = S₄ + S₃ σ,⁽⁴> + S₂ σ₂ ⁽⁴> = 0 + cc¹⁰- α⁴ + α¹⁴- α¹² = _α ¹⁰ Remedy(x) = (cL^ / d_p ) x^(μ"p) σ^(p)(x) = (d₄ / d_p ) x² ( 1 + α³ x)

= α¹⁰x²+α¹³x³ Temp(x) = σ^(μ)(x) = σ⁽⁴⁾(x) = 1 + α⁴ x + α¹² x², σ^(μ)(x) = σ⁵⁾(x) - σ^(μ)(x) + Remedy(x) = 1 + α⁴ x + α³ x² + ¹³ x^J since 2-l_μ—4 ≤mu-4, then /„ mu+l- l_μ =3, μ-p <= 1,

σ^(p)(x) <= Temp(x) = 1 + ⁴ x + α¹ 1²2 x _v2

After mu=4, we have:

l = 3 μ-p = 1

σ^(μ)(x)=σ⁽⁵⁾(x) = 1 + α⁴ x + ³ x² + cc¹³ x³ σ^p)(x) = 1 + α⁴x + α¹²x²

Remedy(x) = α¹⁰x²+α¹³x³

Temp(x) = 1 + α⁴ x + cc¹² x² for mu=5: d₅ = S₅ + S₄ σ,⁽⁵⁾ + S₃ σ₂ ⁽⁵⁾ + S₂ σ₃ ⁽⁵⁾ = α¹³

Remedy(x) = (d^/ d_p ) x^(μ"p) σ^(p)(x) = (d₅/ d_p ) x'(l + α⁴ x + α¹² x²)

= ³ x + α⁷ x² + x³ Temp(x) = σ^(μ)(x) = σ⁽⁵⁾(x) = 1 + ⁴ x + α³ x² + α¹³ x³ σ^(μ)(x) = σ⁽⁶⁾(x) «= σ^(μ)(x) + Remedy(x) = 1 + ⁷x + ⁴ x² + ⁶ x³ since 2 i_μ=6 > mu=5, then (μ-p) <= (μ-p)+l = 2.

After mu=5, we have:

l = 3 (unchanged) μ-p = 2 dp = ¹⁰ (unchanged) σ^(μ)(x)=σ⁽⁶⁾(x) = 1 + α⁷x + α⁴ x² + ⁶ x³ σ^(p)(x) = 1 + α⁴ x + ¹² x² (unchanged)

Remedy(x) = α³ x + ¹ x² + x³

Temp(x) = 1 + ⁴ x + α³ x² + ¹³ x³

At the end of iterations, the error location polynomial obtained is given as σ(x) = σ⁽⁶⁾(x) = 1 + α⁷x + α⁴ x² + ⁶ x³, which is identical to the polynomial given in the example in Lin & Costello. However, note that the intermediate result σ⁽⁵⁾(x) is different. Note further that the determination of the error location polynomial in block 204 only depends on the algebra of the Galois field. It does not depend on how the generating polynomial is defined. Thus, for various standards such as DVB and INTELSAT, this step of building the Berlekamp table is identical, that is, no hardware redesign is required.

Block 206: Error Symbol Location Determination The operation in the error symbol location determination block is a trial-and- error process. In particular, the variable x is replaced in the error location polynomial σ(x) by β°, β, β², β³, ..., β^n"' (where n is the block size of the codeword, e.g., n=255 if using GF(256)) to determine if:

Eq.(12) σ(β' ) = 0,

where β is given in Eq.(la). If β¹ =0, then it is determined that an erroneous symbol is located at (n-j)%n (% is the modulo operation). For example, for the INTELSAT standard (β=α⁵³ , GF(256)), if σ(l=β°) = 0, then the 0^th symbol is declared corrupted; if σ(β') = 0, then the 254^th symbol is declared corrupted; if σ(β²⁵⁴) = 0, then the 1^st symbol is declared corrupted and so on.

Block 210: Error Pattern Determination

There are two steps in finding the error patterns at those error locations identified in block 206 above. First, an error evaluator polynomial Z(x) is determined as given in the Lin & Costello reference:

Eq.(13) Z(x) = l + (S₀ + σ₁) x + (S_l + S₀-σ₁ + σ₂) x² + ...

+ (S_v_, + S_v.₂-σ, + S_v_₃- σ₂ + ... +S₀-σ_v., + σ_v)x^v

where σ, _{23 v} are the coefficients of the error location polynomial in Eq.(l 1) from block 204 and S_0>ι_{;2,3„ v} are the syndromes computed in block 200.

After the error evaluator polynomial Z(x) has been computed, the error pattern at each error location can be determined. The error pattern formula is a modified general form based on equation (6.35) disclosed in Lin & Costello.

Eq.(14) e_Jy = β-^J*"'^{) •} Z(β-^J") / π -_y (1+ β^) where j_y is a particular error location for which the error pattern is being determined, all of the error locations are given by j, _{23 y y v} (i.e., v total errors as found in block 204) and β and r are defined in the generating polynomial of Eq.(la). Z(β^"jy) is the value of the error evaluator polynomial Z(x) evaluated with x = β ^J . Note that this modified general form for determining the error pattern at each error location is universally applicable for arbitrary β, r specified in the aforementioned generating polynomial. This is significant in that it enables the decoding of arbitrary Reed- Solomon codes generated by the generating polynomial g(x) given in Eq.(la) using a common hardware structure. Referring to FIG. 5, a flow diagram of the steps used in block 210 (FIG. 2) for finding the error pattern for the error locations is shown. In block 502, the error evaluator polynomial Z(x) is computed using the coefficients of the error location polynomial σ(x) from block 204 (FIG. 2) and syndromes from block 200 (FIG. 2). In block 504, elements β are raised to certain powers corresponding to the error locations. In block 506, the error pattern at a particular error location is computed using Eq.(14).

Block 212: Error Pattern Determination

Once the error pattern at each error location has been determined in block 210, the errors can be corrected simply by flipping the errored bits in each error symbol. This then provides the original codeword that was transmitted.

In summary, an encoding/decoding algorithm of the present invention has been disclosed herein which can be used to design Reed-Solomon codecs to meet various industry specifications such as the DVB and INTELSAT standards as well as emerging cable modem standards. The universal RS coder/decoder framework is provided in such a way that a user only needs to specify four parameters to configure and define the RS coding scheme, and a coder/decoder can be readily available with a minimum amount of modification of such a universal codec. The algorithm further employs two lookup tables to expedite addition and multiplication operations. While this invention has been particularly shown and described with references to preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims

CLAIMSWhat is claimed is:

1. A Reed-Solomon (RS) decoder for decoding a received codeword block represented by a codeword polynomial R(x) encoded using a RS code (n,k,t) and a generating polynomial of the form g(x)= (x+╬▓^r) (x+╬▓^14"1) (x+╬▓¹^²) ...

(x+╬▓^r+2t_1), where n is a number of encoded codewords in the block, k is a number of information symbols, t is a number of correctable erroneous symbols, ╬▓ is an element of a Galois field, r is the initial exponent for ╬▓, and each ╬▓^¹ is a root of g(x), the decoder comprising: a syndrome computer for computing a series of syndromes from the received codeword polynomial R(x); an error location polynomial computer for computing an error location polynomial of degree v < t using the computed syndromes; means for determining error symbol locations by evaluating the error location polynomial for each element in the Galois field; and means for determining an error pattern for each error symbol by 1) computing an error evaluator polynomial Z(x) based on the error location polynomial and the computed syndromes and 2) computing the error pattern at each error symbol location according to the expression: e_jy = p-^j^-'^{) ΓÇó} z(╬▓-^jy) / ╬╣r,₌,,ΓÇ₧_v (i+ ╬▓"-^jy) where j_y is a particular error location for which the error pattern is being determined and Z(╬▓^"j ) is the value of the error evaluator polynomial Z(x) evaluated with x = ╬▓^jy.

2. The Reed-Solomon decoder of Claim 1 further comprising means for correcting errors in each error symbol according to the determined error patterns.

3. The Reed-Solomon decoder of Claim 1 wherein the error location polynomial computer uses an iterative Berlekamp table.

4. A Reed-Solomon (RS) decoder for decoding a received codeword block represented by a codeword polynomial R(x) encoded using a RS code (n,k,t) and a generating polynomial of the form g(x)= (x+╬▓^r) (x+╬▓¹^¹) (x+╬▓^²) ...

(x+╬▓^²'^"1), where n is a number of encoded codewords in the block, k is a number of information symbols, t is a number of correctable erroneous symbols, ╬▓ is an element of a Galois field, r is the initial exponent for ╬▓, and each ╬▓^¹ is a root of g(x), the decoder comprising: means for computing a series of syndromes, each syndrome S, computed according to the expression R(╬▓^r+1) for i ranging from 0 to 2t-l; means for computing an error location polynomial using the computed syndromes SΓÇ₧ the error location polynomial represented by ╧â(x) = 1 + ╧â, x + ╧â₂ x² + ... + ╧â_v_, x^v_1 + ╧â_v x^v where v is the degree of the error location polynomial and v < t; means for determining error symbol locations by evaluating the error location polynomial ╧â(x) for each element ╬▓¹ for i ranging from 0 to n-1; and means for determining an error pattern for each error symbol by 1) computing an error evaluator polynomial represented by Z(x) = 1 + (S₀ + ╧â,) x + (S, + S₀-╧â, + ╧â₂) x² + ... + (S_v-1 + S_v.₂-╧â, + S_v.₃- ╧â₂+ ... +S₀-╧â_v., + ╧â_v )x^v where ╧â_1>2(._.._v are coefficients of the error location polynomial ╧â(x) and So_,╬╣_{,2, v} are the computed syndromes and 2) computing the error pattern at each error symbol location according to the expression: e_jy = ╬▓-^^r-¹⁾ - z(╬▓-^jy) / ╧Çv_1>._y(i+ ╬▓^J,-^jy) where j_y is a particular error location for which the error pattern is being determined and Z(╬▓^"jy) is the value of the error evaluator polynomial Z(x) evaluated with x = ╬▓^~jy.

5. The Reed-Solomon decoder of Claim 4 further comprising means for correcting errors in each error symbol according to the determined error patterns.

6. The Reed-Solomon decoder of Claim 4 wherein the means for determining the error location polynomial uses an iterative Berlekamp table.

7. A method for Reed-Solomon (RS) decoding comprising the steps of: receiving a codeword block represented by a codeword polynomial R(x) encoded using a RS code (n,k,t) and a generating polynomial of the form g(x)= (x+╬▓^r) (x+╬▓^r+1) (x+╬▓^r+2) ... (x+╬▓^r+2t-'), where n is a number of encoded codewords in the block, k is a number of information symbols, t is a number of correctable erroneous symbols, ╬▓ is an element of a Galois field, r is the initial exponent for ╬▓, and each ╬▓ is a root of g(x); computing a series of syndromes, each syndrome S, computed according to the expression R(╬▓^r+1) for i ranging from 0 to 2t-l; computing an error location polynomial using the computed syndromes S_i; the error location polynomial represented by ╧â(x) = 1 + ╧â, x + ╧â₂ x² + ... + ╧â_v_, x^v_1 + ╧â_vx^v where v is the degree of the error location polynomial and v < t; determining error symbol locations by evaluating the error location polynomial ╧â(x) for each element ╬▓¹ for i ranging from 0 to n-1 ; and determining an error pattern for each error symbol by 1) computing an error evaluator polynomial represented by Z(x) = 1 + (S₀ + ╧â,) x + (S, + S₀╬î, + ╧â₂)x²+ ... + (S_v._! + S_v_₂-╧â, + S_v.₃- ╧â₂+ ... +S₀-╧â_v_, + ╧â_v)x^v where ╧â_lj2> ._..v are coefficients of the error location polynomial ╧â(x) and S_{0╬╣l)2╬╣}.._,v are the computed syndromes and 2) computing the error pattern at each error symbol location according to the expression: e_{jy =} ╬▓-J^y(r-ⁱ⁾ . _Z(╬▓-^Jy) / ╧Çv₁,₁,_y(l+ ╬▓^ji-^jy) where j_y is a particular error location for which the error pattern is being determined and Z(╬▓^"jy) is the value of the error evaluator polynomial Z(x) evaluated with x = ╬▓^~Jy.

8. The method of Claim 7 further comprising correcting errors in each error symbol according to the determined error patterns.

9. The method of Claim 7 wherein the step of determining the error location polynomial includes iterating a modified Berlekamp table.