Implementation of Galois Field Arithmetic. Nonbinary BCH Codes and Reed-Solomon Codes
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1 BCH Codes Wireless Information Transmission System La. Institute of Communications Engineering g National Sun Yat-sen University
2 Outline Binary Primitive BCH Codes Decoding of the BCH Codes Implementation of Galois Field Arithmetic Implementation of Error Correction Noninary BCH Codes and Reed-Solomon Codes
3 Preface The Bose, Chaudhuri, and Hocquenghem (BCH codes form a large class of powerful random error-correcting cyclic codes. This class of codes is a remarkale generalization of the Hamming codes for multiple-error correction. Binary BCH codes were discovered y Hocquenghem in 959 and independently y Bose and Chaudhuri in 960. Generalization of the inary BCH codes to codes in p m symols (where p is a prime was otained y Gorenstein and Zierler. Among the noninary BCH codes, the most important suclass is the class of Reed-Solomon (RS codes. Among all the decoding algorithms for BCH codes, Berlekamp s iterative algorithm, and Chien s search algorithm are the most efficient i ones.
4 Description of the BCH Codes Wireless Information Transmission System La. Institute of Communications Engineering g National Sun Yat-sen University
5 Binary Primitive BCH Codes m integer m, t, a inary BCH codes with m n Block length nk mt Numer of parity-check digits dmin t Minimum distance Clearly, this code is capale of correcting any comination of t or fewer errors in a lock of n m digits. We call this code a t-error-correcting BCH code. The generator polynomial of this code is specified in terms of its roots from the Galois field GF( m m Let e a primitive element of GF(. The generator poly. g( of the t-error-correcting BCH code of length m is the lowestdegree poly. over GF( which has t..... as its roots. 5
6 Binary Primitive BCH Codes t It follows from Theorem. that g( has,,..., and their i conjugates as all its roots. Let i ( e the minimal poly. of. Then g( must e the least common multiple (LCM of, (,..., (, that is, ( t g ( LCM (, (,... ( t If i is an even integer, it can e epressed as a product of the following form : l i i', ' l i i where i is an odd numer and l. Then ( is a conjugate ' i i of and therefore and i have the same minimal poly., that is, ( (. i i 6
7 Binary Primitive BCH Codes Hence, even power of has the same minimal poly. as some preceding odd power of. deg ( deg ( g ( LCM (, (,... t ( i m (Theorem.8 &.9 g mt n k mt The BCH codes defined aove are usually called primitive (or narrow-sense BCH codes. 7
8 Binary Primitive BCH Codes The single-error-correcting BCH codes of length m is generated y g ( ( since t =. m is a primitive element of GF( ( is a primitive poly. of degree m 0 m (,,..., the single-error-correcting BCH codes of length Hamming code E 6. m is a 4 is a primitive element of GF( given y Tale.8 such that 4 0. The minimal polynomials of α,α,α 5 are 8
9 Binary Primitive BCH Codes ( 4 4 ( ( 5 5 The doule-error-correcting BCH code of length n 4 5 is generated y g( LCM{ (, ( } Since and ( are two distinct irreducile polynomials, ( g( ( ( (5,7 cyclic code with d min W ( g ( 5 d 5 min 4 9
10 Binary Primitive BCH Codes The triple-error-correcting BCH code of length 5 is generated y g ( LCM (, (, ( ( ( ( It s a (5,5 cyclic code with dmin 7.Since W( g( 7, dmin 7. Let v ( v0 v... v n n e a poly. t t with v GF( If v( has roots,,...,, then v( is divisile y (, (,..., (. (Theorem.4 t i v( is a code poly. ecause ( g( LCM{ (, (,..., ( } v t 0
11 Binary Primitive BCH Codes We have a new definition for t-error-correcting BCH code: Ainaryn-tuple n v ( v 0, v, v,..., v n is a code word if and only if the poly. ( i.e. v( i v i i v v... v n i 0 n 0 i. v, v, v,..., v n 0 for i.. ( n i ( 0 t.
12 Binary Primitive BCH Codes Binary Primitive BCH Codes Let... n.. (... (... n.. H If v is a code word in the t error correcting BCH code (... (.. n t t If v is a code word in the t-error-correcting BCH code, then 0 H v T The code is the null space of the matri H and H is the parity check matri of the code 0 H v parity-check matri of the code.
13 Binary Primitive BCH Codes j i j i is a conjugate of, then v( 0 iff v( 0 j-th row of H can e omitted. As a result H can e reduced to the following form : H n ( ( n t t t n ( ( ( (Thm.. 只剩下奇數的 Row EX 6. doule-error-correcting BCH code of length n 4 5, (5,7 4 code. Let e a primitive element in GF(
14 Binary Primitive BCH Codes The parity-check matri is (Note that the parity check matri is not inary anymore H Using 5, and representing each entry of H y its 4-tuple, H
15 Binary Primitive BCH Codes FACT: The t-error-correcting BCH code indeed has dmin t (pf: Suppose v 0 such that W(v t Let v, v,... v e the nonzero components of v (i.e. all ones j j j T ( ( 0 vh ( v jv.. j v j t ( ( j j t j ( ( j j t j j j j j j j t ( ( j j j t ( ( (,..., 0... j j j t ( ( 5
16 Binary Primitive BCH Codes j j j ( ( j j j ( ( (,..., 0 j j j ( ( A, a square matri t A 0 Taking out the common j ( j factor from each row of j ( j the determinant. ( j j... j 0... ( j j 6
17 Binary Primitive BCH Codes The determinant in the equality aove is a Vandermonde determinant which is nonzero. The product on the left-hand side of can not e zero. This is a contradiction and hence our assumption that there eists a nonzero code vector v of W(v t is invalid. dmin t t+ is the designed ddistance of fthe t-error-correcting ti BCH code. The true minimum distance of a BCH code may or may not e equal to its designed distance. m Binary BCH code with length n can e constructed in the same manner as for the case n m. (Theorem.9 m m Let e an element of order n in GF(, n and g( e the t inary polynomial l of minimum i degree that t has,,..., as roots. 7
18 Binary Primitive BCH Codes Let (, (,..., t ( e the minimal poly. of respectively then g( LCM{ (, (,..., t ( } n t n,,,..., are roots of n g ( ( X We see that g( is a factor of +. n,,... t The cyclic code generated y g( is a t-error-correcting BCH code of length n. The numer of parity-check digits dmin t. If βis not a primitive element of GF( m, the code is called a nonprimitive BCH code. mt 8
19 General Definition of Binary BCH Codes General definition of inary BCH codes. GF( m, 0 e any nonnegative integer and consider l0 l0 l0 d 0,,...,. For 0 id0, let i (, ni e the 0 i minimal poly. and order of, respectively. g( LCM{ 0(, (,..., d ( } and dthe length of fthe code is Note that: dmin d0 n LCM n, n,... n d 0 (Proof is omitted and left as an eercise. parity-check digits md ( 0 is capale of correcting ( d0 / or fewer errors 0 0 9
20 General Definition of Binary BCH Codes If we let l0 =, d0 = t+ and e a primitive element of GF( m, the code ecomes a t-error-correcting primitive BCH code of length m. If we let l0 =, d0 = t+ and e not a primitive element of GF( m, the code is a nonprimitive t-error-correcting BCH code of length n, which is the order of. For a BCH code with designed distance d0, we require g( has d0 - consecutive powers of a field element as roots. This guarantees that the code has d. This lower ound on the minimum min d 0 distance is called the BCH ound. In the rest of this chapter, we consider only the primitive BCH codes. 0
21 Decoding of the BCH Codes Wireless Information Transmission System La. Institute of Communications Engineering g National Sun Yat-sen University
22 Decoding of the BCH Codes Decoding of the BCH Codes Suppose that a code word v( = is transmitted and the transmission errors result : 0... n v n v v v is transmitted and the transmission errors result : r( = Let e( e the error pattern. Then 0... n r n r r r For decoding, rememer e( v( r( ( (... n n.. (... ( H n ( (.... n t t (... ( n t t
23 Decoding of the BCH Codes The syndrome is t-tuple, S ( S, S,... S t r H i i i n Let s r ( r r... r ( for i i 0 n s can e evaluated y ( R [ r( ] i i ( i ( is the minimal poly. of i r( a ( ( ( s r i i i i i i ( i( i T t
24 Decoding of the BCH Codes EX6.4 Consider the doule-error-correcting error (5, 7 BCH code given in 8 (from E6.. If r ( r( s (, s ( 4 4 ( ( 4 ( ( 4 ( R [ r ( ] ( ( ( R [ r( ] s s ( 4 9 ( s (,,, (Note that syndrome is not inary anymore. 4
25 Decoding of the BCH Codes i i i v( 0 for i t s r( e( j j j Suppose e (... v 0 j j... j n j j jv s... j j j s ( (... ( v i j t jv t s tt (... ( j j j where,,... v are unknown. Any method for solving these equations is a decoding algorithm for the BCH codes. v 5
26 Decoding of the BCH Codes j j j Once,,... v have een found, the powers j, j,... j tell us the error locations in e( v If the numer of errors in e( is t or less, the solution that yields an error pattern with the smallest numer of errors is the right solution. j For convenience, let, v e the error location numers. s... v s... v st... t t t v Power-sum Symmetric function 6
27 Decoding of the BCH Codes Define ( ( (...( = 0... : error-location poly. The roots of ( are,,... v, which are the inverses of the error location numers v... v v v v v v v 7
28 Decoding of the BCH Codes These coefficients are known as elementary symmetric functions. i ' s are related to s ' s y Newton's identities s j 0 s s 0 ss s 0 sv sv... v svv 0 s s... s s 0 v v v v Note that, for inary case, 0, we have i for odd i i i 0 for even i 8
29 Decoding of the BCH Codes The equations may have many solutions. We want to find the solution that yields a σ(x ofminimal degree. This σ(x would produce an error pattern with minimum numer of errors. Decoding Procedure step. Compute s ( s, s,... s from r ( t step. Determine ( from step. Determine the error-location numer,,... v y finding the roots of ( and correct the errors in r ( s 9
30 Decoding of the BCH Codes Iterative Algorithm for finding ( (Berlekamps iterative algorithm i i. e ( i s s Y i t ( t ( ( N END D o es the coeficients ( i of satisfy the ( i th N e w to n s id en tity? Y ( i ( i correction term N ( i ( i a i i 0
31 Decoding of the BCH Codes How to add a correction term to ( i? ( ( ( ( Let... e the minimum-degree poly. determined at the thstep. ( To determine ( => compute th discrepancy ( ( (... d s s s s ( ( If d 0 ( ( If d 0, go ack to the steps prior to the th step and determine ( ( s.t. th discrepancy d 0, and has the largest value. ( ( deg [ ( ]. Then ( ( ( ( ( ( d d ( is the minimum-degree poly. whose coefficients satisfy the first Newton s identities
32 Decoding of the BCH Codes To carry out the iteration of finding σ(x, we egin with the following tale: ( ( d 0-0 S 0 0 t
33 Decoding of the BCH Codes l μ is the degree of σ (μ (X. If d μ =0, then σ (μ+ μ (X= σ (μ (X andl l μ+ = l μ. If d μ 0, find another row ρ prior to the μth row such that d ρ 0 and the numer ρ-l ρ in the last column of the tale has the largest value. Then σ (μ+ (X is given y ( ( d d ( ( ( ( ( and l μ+ = ma(l μ, l ρ +μ-ρ ρ. ( ( In either case, d s s... s (t The polynomial σ (X in the last row should e the required σ(x.
34 Decoding of the BCH Codes E 6.5 Consider (5.5 5 triple-error-correcting error BCH codes given in e 6. (t= p ( e v 0 r ( ( ( ( ( (
35 Decoding of the BCH Codes ( R [ r ( ] ( ( ( R [ r( ] ( R [ r( ] 5 5 ( 4 s s s s s 6 s s (,,,,, ( ( ( ( ( ( d d ( 5
36 Decoding of the BCH Codes S = ( ( (... μ=0; d 0 =S = 0 ( ( ( ( ρ=- ( ( d d ( σ ( (X= σ (0 (X+d 0 d - - X (0+ σ (- (X=+ X =+X X +X l =ma(l 0,l - +μ-ρ=ma(0,0+0+= μ-ll μ =-l l =-=0 d s s s s d =S +σ ( S =+ =0 l μ+ = ma(l μ, l ρ +μ-ρ. σ ( (X= σ ( (X=+X l =l = μ-l μ =-l =-= d =S +σ ( S =α 0 + =(+α+α += α 5 6
37 Decoding of the BCH Codes d =α 5 0 ρ=0 σ ( (X=σ ( (X+d d 0 - X (-0 σ (0 (X=+X+ α 5 X =+X+α 5 X l =ma(l,ll 0 +μ-ρ=ma(,0+-0= 0 μ-l μ =-l =-= d S 5 =S 4 +σ ( S +σ ( S +σ ( S =+ α 0 + α 5 +0 =+(+α+α +(α+α =0 7
38 Decoding of o f the t h e BCH Codeso d e s ( ( d ( ( (6 5 0 ( ( (,, 0 8
39 Decoding of the BCH Codes error location numers,, e ( r ( r ( e ( 0 If the numer of errors in the received polynomial r(x is less than the designed error-correcting capaility t of the code, it is not necessary to carry out the t steps of iteration to find the errorlocation polynomial σ(x. It has een shown that if d μ and the discrepancies at the net t-l μ - steps are all zeros (i.e. successive t-l μ zeros, σ (μ (X is the errorlocation polynomial. If v(v t errors occur, only v+t steps of iteration are needed. The iterative algorithm descried aove not only applies to inary BCH codes ut also noninary BCH codes. 9
40 Decoding of the BCH Codes Simplified Algorithm for finding ( For a inary BCH code, it is only required to fill out a tale with t empty rows. Such a tale is presented elow. ( ( d -/ 0-0 S t 40
41 Decoding of the BCH Codes ( (. If d 0, then ( (. If d 0, find another row proceding the th row, s.t. is as large as possile and d 0 ( ( ( ( Then ( X ( X d d X ( X ( ( note that d s s s +... ( s ( deg [ ( ] 4
42 Decoding of the BCH Codes The polynomial σ (t (X in the last row should e the required σ(x. If it has degree greater than t, there were more than t errors, and generally it is not possile to locate them. The computation required in this simplified algorithm is one-half of the computation required in the general algorithm. The simplified algorithm applies only to inary BCH codes. If the numer of errors in the received polynomial r(x is less than the designed error-correcting capaility t of the code, it is not necessary to carry out the t steps of iteration to find the errorlocation polynomial σ(x for a t-error-correcting inary BCH code. 4
43 Decoding of the BCH Codes Remarks: If v t errors occur, only ( t v / steps needed. If, for some μ, d μ and the discrepancies at the net (t-l μ -/ steps are zero, then ( X is the error-location poly. EX6.6 The simplified tale for finding ( for the code in e6.5 is given elow. Thus, ( ( ( 5. ( ( d I I -/ 0-0 S= 0 0 +S=+ 5 S+S S== (take = -/ (take = (take = 4
44 Decoding of the BCH Codes Finding the Error-Location Numers and Error Correction. Consider e The error-location poly. has een found to e 5 ( By sustituting 4 0,,,...,, into it, we find that,, are the roots of (. Therefore, the error location numers are 5 5,, e( Chien s procedure: The received vector r( r rn 0 r r... is decoded on a it-y-it asis. The high-order its are decoded first. n To decode r n-, the decoder test whether is an error-location numer; this is equivalent to test whether its inverse is a root of (. If is a root, then 44 n v... v 0
45 Decoding of the BCH Codes l l vl To decode rn l, the decoder forms,,..., v and tests the sum l l... If the sum is zero, then α n-l is an error-location numer and r n-l is an erroneous digit; otherwise, r n-l is a correct digit. Cyclic error location search unit v vl t t i i A i t 45
46 Decoding of the BCH Codes The t -registers are initial stored with,,..., t calculated in step of the decoding ( v v... t 0 for v < t. Immediately efore rn- is read out of the uffer, the t multiplier are pulsed once. v The multiplications are performed and,,..., v are stored in the -registers. The output of the logic circuit A is if v and only if the sum otherwise, the... v 0; output tof fa is 0. The digit itrn- is read out of fthe uffer and corrected y the output of A. Having decoded rn-,the t multipliers are pulsed 4 v again. Now,,..., v arestoredinthe the -registers. The sum 4 v... v is tested for 0. The digit rn- is read out of the uffer and corrected in the same manner as rn- is corrected. 46
47 Implementation of Galois Field Arithmetic Wireless Information Transmission System La. Institute of Communications Engineering g National Sun Yat-sen University
48 Implementation of Galois Field Arithmetic Galois field adder (GF( 4 given y Tale.8 ADD a0 a a a Register A (accumulator a0 0 a 0 a a Register B By adding their vector representations with modulo- addition. 48
49 Implementation of Galois Field Arithmetic For multiplication, we first consider multiplying a field element y a fied element from the same field. Suppose that we want to multiply a field element in the GF( 4 y the primitive element 4 whose minimal polynomial is (. 0 ( 0 任一 β 要乘上定值 α 4 0 E: 7 let. The vector representation of is ( 0. Load it into the register. After the register is pulsed, the new content in it will e (
50 Implementation of Galois Field Arithmetic the aove circuit can e used to generate all the nonzero element 4 GF( y load (000 first into the register E: as another eample, multiply β y α : ( ( + ( ( ( (
51 Implementation of Galois Field Arithmetic E: multiplying two aritrary elements Steps: 0 r c c c c 0 r c c c c 0 ((( c c c c 0. multiply c y and add the product to c. multiply ( c c y and add the product to c. multiply (( c c c y and add the product to c 0 5
52 Implementation of Galois Field Arithmetic The circuit in e can e modified to carry out the computation given y e 0 c0 c c c 5
53 Implementation of Galois Field Arithmetic Step : Register A are empty B ( 0 C ( c0 c c c Step : Registers A. C. are shifted four times. At the end of first shift, A (( c 0, c, c, c, the vector representation of c At the end of second shift, A (( c c At the end of third shift, A ((( c c c At the end of fourth shift, A 5
54 Implementation of Galois Field Arithmetic If we epress ((( c c 0 c c then we otain a different multiplication circuit. 0 c c c c0 54
55 Implementation of Galois Field Arithmetic How to compute syndrome It can e done with a circuit for multiplying a field element y α i. Suppose that we want to compute si i r(? where α is a primitive element in GF( 4 r( r r r... r =(...((( r r r... r
56 Implementation of Galois Field Arithmetic The computation of r(α can e accomplished y adding an input to the circuit r ( input At the end of first shift ( r,0,0, At the end of nd shift r 4 r At the end of rd shift ( r4 r r 56
57 Implementation of Galois Field Arithmetic We can compute r( y adding an input to the circuit for multiplying y of figure in E. r ( input i There is another way of computing r(. Let i ( e the minimal i poly. of. Let ( e the remainder resulting from dividing r( y i (. Then i i r( ( 57
58 Implementation of Galois Field Arithmetic Implementation of Galois Field Arithmetic Thus, computing r( is equivalent to computing (. Again we consider computation over GF( Suppose that we want to i i 4 consider computation over GF(. Suppose that we want to compute r(. The minimal poly. of is 4 ( The remainder ( ( Then 0 ( ( ( 0 0 ( ( ( 58
59 Implementation of Galois Field Arithmetic Such a circuit is shown elow 59
60 Implementation of Galois Field Arithmetic Implementation of Galois Field Arithmetic Since is a conjugate of, it has the same minimum poly. as 6 6 ( ( 0 0 ( ( ( ( ( Its circuit is shown in the net page 0 ( ( ( 60
61 Implementation of Galois Field Arithmetic 6
62 Implementation of Error Correction Wireless Information Transmission System La. Institute of Communications Engineering g National Sun Yat-sen University
63 Implementation of Error Correction Syndrome computation circuit for the doule-error-correcting (5,7 BCH code 6
64 Implementation of Error Correction Chien s searching circuit for the doule-error correcting (5,7 BCH code. 64
65 Noninary BCH Codes and Reed-Solomon Codes Wireless Information Transmission System La. Institute of Communications Engineering g National Sun Yat-sen University
66 Noninary BCH Codes and Reed-Solomon Codes In addition to the inary codes, there are noninary codes. If p is a prime numer and q is any power of p (q=p p m thereare, codes with symols from the Galois field GF(q. These codes are called q-ary codes. An (n, k linear code with symols from GF(q is a k-dimensional suspace of the vector space of all n-tuples over GF(q. A q-ary (n, k cyclic code is generated y a polynomial of degree (nk with coefficients from GF(q, which is a factor of X n -. Encoding and decoding of q-ary y codes are similar to that of inary codes. For any choice of positive integers s and t, there eists a q-ary BCH code of length n=q s -, which can correct any cominational of t or fewer errors and requires no more than st parity-check digits. 66
67 Noninary BCH Codes and Reed-Solomon Codes Let α e a primitive element in the Galois field GF(q s. The generator polynomial g(x ofat t-error-correcting q-ary BCH is the polynomial of lowest degree with coefficients from GF(q for which α,, α,,, α t are roots. Let φ i (X e the minimal polynomial of α i. Then g(x=lcm{φ (X,φ (X,,φ t (X } The degree of each minimal polynomial is s or less. Therefore, the degree of g(xisatmostst st, and hence the numer of parity-check digits of the code generated y g(x is no more than st. For q=,, we otain the inary BCH codes. 67
68 Noninary BCH Codes and Reed-Solomon Codes t-error-correcting Reed-Solomon code with symols from GF(q: s =, Block length: n=q-, # parity-check digits: n-k =t, Minimum distance: d min =t+. Reed-Solomon codes with code symols from GF( m (ie (i.e., q= m Let α e a primitive element in GF( m The generator polynomial of code length m -is g( ( (...( = g0 g... gt (n,n-t n cyclic code and coefficients of g(xisfromgf( m k Let Message: a X a0 ax ax ak X In systematic form, the t parity-check digits are coefficients of t t t t t X Rg X X a X 0 X t X (Sec 5. 68
69 Noninary BCH Codes and Reed-Solomon Codes Encoding circuit of a q-ary RS code 69
70 Noninary BCH Codes and Reed-Solomon Codes Decoding of RS code: v(= = r( = v0 v... vn r 0 r... r n n n e ( r ( v ( ee... e 0 n n e ( v( r ( m ei ri vi, GF( Suppose the error pattern e( ( contains v errors (nonzero j components, then j j e( e e... e j j j j To determine e(, we need error-locations l s and the error jl values (i.e. vpairs(, e ' s j l In decoding a RS code, the same three steps used for decoding a inary BCH code are required ; in addition, a fourth step involving i calculation of error values is required. 70
71 Noninary BCH Codes and Reed-Solomon Codes j Let... v s r( e e... e j j jv v s r ( e... e j jv v s r( e.. e t t t t j jv v note: S R [ r( ] i i ( i ( ie.. r ( a ( ( i i i 7
72 Noninary BCH Codes and ReedReed-Solomon Codes Syndrome computation circuit for R-S codes Division circuit (a over GF ( m ( in inary form 7
73 Noninary BCH Codes and Reed-Solomon Codes To find ( : the error-location poly. y Berlekamp s iterative algorithm ( ( (...( v... v Once ( is found, we can determine the error values. Let... z s s s s s v v j It can e shown that the error value at location l is given y: e j l i il Z l i l l 7
74 Noninary BCH Codes and Reed-Solomon Codes E: Consider a triple-error-correcting Reed-Solomon code with symols from GF( 4 The generator polynomial of this code is: g( ( (...( (5, 9, RS code, n =5. n k = 6, t = e ( r ( r (
75 Noninary BCH Codes and ReedReed-Solomon Codes Step. Using tale.8 elow to compute syndrome components: 75
76 Noninary BCH Codes and Reed-Solomon Codes S r( S S S S S r ( r ( r( r ( 0 r( Step. Find( y filling out tale in net slide. 76
77 Noninary BCH Codes and Reed-Solomon Codes 77
78 Noninary BCH Codes and Reed-Solomon Codes Step. By sustituting 9 4,,,... into (,, are roots of ( 9 6,, are the error-location 6 numers of e ( e ( e e e 6 Step 4. From 6.4, find z( z( 6 7 e 6 ( ( e 6 z ( ( ( 78
79 Noninary BCH Codes and Reed-Solomon Codes e z( ( ( 6 4 e ( v ( r ( e ( 0 If β is not a primitive iti element of GF( m, then the m -ary code generated y t g ( ( (...( is a nonprimitive t-error-correcting RS code. The length n of this code is simply pythe order of β. Decoding of a nonprimitive Reed-Solomon code is identical to the decoding of a primitive Reed-Solomon code. 79
Implementation of Galois Field Arithmetic. Nonbinary BCH Codes and Reed-Solomon Codes
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