Vol. 1 No. 1 J O U R N A L OF ELECTRONICS Jan. 1984 , , , ,

A N A L G E B R A I C C O M P L E T E D E C O D I N G F O R

D O U B L E - E R R O R - C O R R E C T I N G B I N A R Y B C H C O D E S

Feng O u ~ a n g ( ~ ~ ~ )

(Shanghai Institute of Computer Technology)

Atmract

An algebraic complete decoding for double-error-correcting binary BCH codes of primitive length i s dtxived. The decoding is done more quickly than the step--by-step decoding devised by Hartmann. And if an error pattern corresponding with s y n d r ~ s 2 and st has weight 3, the decoding can find all error patterns of weight 3 corresponding with these syndromes. At the same time, a discriminant for a polynomial of degree 3 over GF(2 m) has three distinct roots in GF(2 rn) is also derived. The discrimi- nant is very important for complete decoding of triple-error-correcting binary BCH codes.

I. Introduction

From [1] and [2], it is known that the double-error-correcting binary BCH codes of

length n = 2 ' ~ - - I are quasi-perfect, i.e. any coset of the codes contains a vector of weight

3 or less. Therefore all error patterns of two or less errors and some error patterns o f ih ree

en 'ors can be corrected, and there is not any error pattern of four or more errors. These

codes are very useful in data transmission. The s tep-by-s tep decoding for the codes was

devised by HartmanntSL This decoding requires more time when m is large and it has

to be implemented by software. In addition, when any error pattern conta/ns three errors,

the decoding can only find one error pattern. In this paper an algebraical method of the

complete decoding for the codes is derived. The method can be implemented easily and

can find all error patterns o f three errors corresponding with the same syndromes to choose.

At the same time, a discriminant for a polynomial o f degree 3 over GF (2") having three

distinct roots in GF(2 m) is also derived. This is the key problem, which was not solved

in [4].

Let

If. Discriminant for a Polynomial of Degree 3 over,

GF(2-) Has Three Distinct Roots in GF(2m,)

f(x)=a~+a~ +alx~ + ~ (1) be a polynomial of degree 3 over GF(2"). Obviously, i f f (x) is .prime to f'(x), f(x) has no repeated roots. F o r convenience, we always s u p p o s e t ~ t ' f (x ) 1ms no repeated roots.

F rom Kronecker Theorem~6~ about zero points of a polynomial it is known that f ( x ) has

three distinct roots in some extended field GF(2~) , and x~, xl, xa denote the three distinct

roots, we have:

No. 1 AN ALGEBRAIC COMPLETE DECODING 13

{ o']=x~Wx~+xs, cr2 = XxX~ W XaXs + X2Xs, (2) tra =XlX2Xs.

We can get a sequence {sj} by the following recurrence formula: ~%s~ +cr2sj +l+crxs~ +~ =s~ +a. (3)

From the theory of sequence, we have sj=clx~-l-c2x~+cax ~ for j = l , 2, 3, ..., (4)

where ca, ca and ca are dependent on sa, s2 and sa. Lemma 1. If f sa=cza,

s,=~L (5) s3--cr ~ +cr~cr2+ u3,

then Cx=C~=c3=l. Proof. From Eq. (5) and Eq. (2), we obtain

( sa=xa+ x2+ x3, _ _ ~. 2 2 s~ -xa +x2 + x 3, (5') _ _ 3 3 3 ss--xl + x2 + x.~.

Substituting the above equation into Eq. (4), we have

(ca+ 1).xa+(c2+ 1). x2 + (ca+ 1).Xa =0, (ca+ 1)- x~ + (c2+ 1), x[ + (ca+ 1). x~ : 0 , (6) (c~+ 1).x~ +(c~+ l) .x~+(ca+ ]).x~ =0.

Since xa, x~ and xs are distinct from each other, ca+l = c 2 + l = c a + l = 0 , namely c a : % = cs=l . Q~E.D.

In this case, Eq. (4) becomes

sa=xj+x~+x~. (4') Theorem 1. x a, x s and xa belong to GF(2 m) if and only if

{ s~'-I § (7) 3 ' ~ m - - 2 § = S 5 .

Proof. Obviously, if xa, x 2 and x 3 belong to GF (2"), then Eq; (7) is true. Conversely, if Eq. (7) is true, since sa E GF (2"),

s,- =st" =Sl. (7') From Eq. (4'), Eq. (7) and Eq. (7'), we have

( xa (x~m-a+ 1) + x,(xl=-a+ 1) + x~ (x~"-~+ D=0, 3 g m - I 3 2 m a 3 2 m 1 - - ~ xl (xa + l ) + x 2 (x2 - + 1 ) + xs(x 3 - +1)--0 ,

w ' ~ m - I 5 2 m - I 5 2 m - 1 _ _ /xa (xa + 1)+xz (x 2 + l ) + x a ( x s +1)--0.

Since xl, x2, x3 are distinct from each other, x] ~-a + 1 = x] '~-1 + 1 =x~ m-a+ 1 =0, namely x a, x~ and x a belong to GF(2=). Q.E.D.

Lemma 2. If the sequence {s~} satisfies Eq. (3)and Eq. (4'), then the sequence {sj �9 § } ~.o.a,~.... satisfies

2 p . 2 P P cr ~ �9 s~* - f -~ �9 s~ * +~ + ~r a a �9 s~ * +~. ~ =s~. +~.~, (8) where ]* is any integer, p is a non negative integer.

14 JOURNAL OF ELECTRONICS 1984

then

where f~(B) satisfies

Proof. Since the sequence satisfies Eq. (3) and Eq. (4'), we have

~ .s ~.+o-~'.s~*+,d, + ~% �9 ~*+~.,~, + s ~*+s.~,

=-~_ x~ ~ l+tr| ' , x$" +o'~'. x~"" + x ~ ' )

= ' " ( O ' . + ~ ' X , + O ' l ~ . , . . . , 1 .

Since ~ , x~ and xs are three roots of Polynomial (1), the right side of the above equation is equal to zero. Q.E.D.

In order to find a general expression of s ~ +x, we suppose crib0, crs+~r~:~0. Lemma 3. Let

~_, [try=a, o' , / , r I =b,

(s,, +1) / (~rx~ ' +~ )~=A,, (9)

(a+b) ~ / (b+ 1) ~ =B,

Ai-~ (a+b) ( b + l ) ' ' -~ . f , (B)+ 1, (10)

f , (B)--l , A (B) = B + 1, (11)

f ~ (B) = B 2 ' -So f ,_ , (B)..}_f t_ 1 (B).

Proof. We shall use mathematical induction to prove the lemma. Obviously, when i=2 , 3, Eq. (10) is true. Suppose Eq. (10) is true for i--I and i--2. In Eq. (8)letp--i--2, /*=2~-2+1, 1 respectively, we have

2'-~ *'~ ~-~ (12) s ~ + l = o ' s �9 S,~-2+I+Gr~ �9 S2.,~-2,X+U~ "S,.~-%a,

= �9 �9 s,i-%1-t-crx "s~.2~-%1. 03)

:Substituting Eq. (13) into Eq. (12), we obtain

�9 �9 ' " 2 t ~ - - g - - , i - - 1 2 1 " : ' -~,'+1 = (6~ ' - ' + ~r~ '-2 ~ - ' ) " S, '- '+~+t~, -t-~1 ) . S,'-%~+o's �9 Crl~-'-s~.

Dividing both sides by ~r~ ~+1 and using Eq. (9) and s l ~ 1, we have

Ai = (aq-b) '~ e. A~_, q- ( b + l ) ' ~ - ' . A~_~ + a 2~-' . (14)

Since Eq. (10) is true for i - 1 , i--2, substituting them into the above equation, we obtain

A~ = (a+b)"-"[(a+b) (b+ 1)0-'-~-~. f r 1]

+ (b+ 1) '~ ~[ (a+ b) (b+ 1), ~-2~. f~- l (~)+ 1]+ a~ ~- ~

=(a+b) (b+ l ) , , -~_a [ (o+b)~ , - , / ( b+ l )~ . ,~-s f ~_,(B)+f ,_x(B)]+l. Using Eq. (9) and Eq. (11), we have

A~ = (a+b) (b+l) t ' -X-~o f i (B)+ 1, ~amely, Eq. (10) is true for i. So Eq. (10) is true for 2 or more. Q.E.D.

Theorem 2. A discriminant for ~ a - ~ - ~ , x + ~ . x ~ + x ~ = 0 has three distir~ct roots in ~GF(2 m) and is

1'4o. 1 AN ALGEBRAIC COMPLETE DECODING 15

rj2m - 2 4" 1, -Jm-1 (B)+l=O, (15) n 3 S m - ~ �9 �9 1,_~ (B)+ 1 ~0.

Proof, From Theorem 1 it is known:that ~ s + a ~ . x + ~ q . x 2 + ~ , = ~ has three distinct ~roots in GF(2 =) if and only if Eq. ('/).iS true. And,from Eq. ,(9) and Lemma 3, we have

A 2 L-.-d " ' m - I - - ~ 1 , _ (16)

A=_2--Aa.

From Eq. (10), the above equation can be written as

t (a+b)' (b§ 1) ' ' -1 - ' . f~_1 (B)+I = (a+b)+ 1, (16') (a+b) 4 ( b + l ) " - l - ~ . f ~_, (B)+ l=(a+b):(b+ l)_+ l.

And using the following

(a+b) (bd-l) ~ ' - a - s = (a+b)2"] (b+l)a.~m-t = B ~"-l,

(a+b)3 ( b + l ) ~ " - l - s = (a_l_b)Z~§ / (b_}_l)a.2~-l,S~B~"-l .t=_ B 2"-1 +2",

-we have

t B'm-x.f~_t ( B ) + I = 0 ,

B2~-a~2~. r4.,,,,_~ (B)+ 1 =0. Q.E.D. O J L

Since the coefficients off2 (B) and f~(~) axe zero or . l , and from Eq. (11), the coeffi- ~eients of the two eq ,uations in Eq;.l(15) are zero or, 1. Therefore if a is a root of Eq, (15), then.the conjugate elements a ~ are also roots o f Eq. (15). Let g,~(B) denote the minimum polynomial of a. ff all roots of Eq. (15) are in the conjugate classes of a~, a~, ..., a#, then Eq. (15) is equivalent to

g.~ (B) - g~. (~)...,g~ (B)=0. (17)

Theorem 3. Eq. (17)is a diserimirmnt for Polynomial (1) to have three roots in

For example, we consider a polynomial of degree 3 over GF(25). From Eq. (tt), f d ~ = B + l , .fdB):--BZ+B+l. In ttiis ease, "Eq. (15)' becomes

B s ( ~ + B+1)+1=0, B~'. (t~+ l )+ 1=0. (18)

<)bviousl~, o~, a a~ a ~~ a ~ and ~a8 are roots of Eq. (1.8), where a is a primitive element of GF(2n). The minimum polynomial of a 5 is B~+B~+B +B-}-I =0. So it is a discriminant for a polynomial of degree 3 over GF(20 to have three distinct roots in GF(20, where B is obtained from Eq. (9).

When cry=0, it can be checked that Eq. (17) is also a discriminant for Polynomial (1) to have three distinct roots in GF(2"), where B=o'2]o].

When cr2+cr~t :=0, let x=y+~r~, then Polynomial (1) becomes

(o'a+o't~)+f =0. (19)

We can discriminate whether Eq. (19) has three distinct roots in GF(Tn).

16 JOURNAL OF ELECTRONICS t984

IH. Algebraic Complete Decoding

Complete decoding for double-error-correcting binary, BCH codes, for any syndromes. s a, s~, which are not zero simultaneously, is" to find the smallest positive integer i and xj E GF(2~)--{0}, for j = l , 2 . . . . . i, such that

i / (20)

Since the double-error-correcting binary BCH codes of lerigth 2m--1 are quasi-perfect~ Eq. (20) must have solution of i ~ 3. And when i = 1 (or 2) Eq. (20) has unique solution x t (or x I and x~), when i = 3 it has many solutions xt, x= and x.~. Algebraic complete decoding is to find all solutions of Eq. (20) by algebraic method from the syndromes sl and s3. It can be checked easily that:

(1) If s x ~ 0 and A ~ [ sl s= I -- 0, where s~ = Sl 2, then Eq. (20) has unique solution: SZ SS

i = l and x I = s~. rt l-- ~.

(2) I f s 1 ~ 0, A :~ 0 and tr(A[sl) = 0, where tr(fl) zx j~__0/32 J, then Eq. (20) has unique

solution i = 2, and x~, x~ are two distinct roots of the following equation: xa+sz x + (szS+ss) [ sa=O. (21k

Since tr(d/s~) = 0, from The6rem 6,695 in [b-] if is 'known that Eq. (21) has two distinct roots in GF (2=). And zJ % 0, so xj % 0. FrOm the relation between roots and coefficients, it is known that xx and xa satisfy Eq. (20).

(3) If

s t : 0 and .r a ~ 0 (22) o r

s 1 :~: 0, d :~ 0 and tr(A/sl) ~- O, (22') then Eq. (20) has solution i = 3. In this case the key problem is to find xx, x2, xa E GF(2=) - {0} such that Eq. (20) holds. We shall show under the condition (22) or (22') how to find x~, x=, x3 E GF(2 '~) -- {0}, such that

{ xt+x=+x3=st' (20') 3 S 3 _ _ xl + x z + x 3 --s3.

Suppose the solutions of Eq. (20') are three distinct roots of Polynomial (1). From the relation between roots and coefficients, we have

{ ~ =s l ' (23) cr,=~+s~+st, o-=.

In this case Polynomial (1) becomes

(s3+ s] + a x �9 ora)+o'=, x + s 1. x2 + x ~ =0. (24)

To find the roots of Eq. (20') b e ~ m e s the problem of how to choose a2 such that Eq. (24) has three distinct roots in GF(2m)--{0} and how to find the roots of Eq. (24). Under the condition (22) or (22'), it is known easily that Eq. (24) has no repeated roots and x ~ O .

No. 1 AN ALGEBRAIC COMPLETE DECODING 17

From Eq. (24) and Eq. (9), B=(ss+sl)'/(cr,+s~)s. (25)

From Theorem 3, Eq. (24) has three distinct roots in GF (2 m) if and only if aj =(s .+sD' / (~+s~ ) ; (26)

namely,

cr~ =[(s~ +s~) [ all x tsWs], (26') where a~ is a root of Eq. (17), or the condition (22) holds, then

trs =0, (27) or the condition (22') holds, then

r =0. (28) In the ease of Eq. (27), Eq. (24) becomes s , + x a = 0 , if and only if m is even and ss-----fl *t,

then Ecl. (24) has three distinct roots in GF(2=), where/3 is a primitive element of GF(2=). In the ease of Eq. (28), Eq. (24) becomes ss+s~.x+sx.xS+xa=O. Let x=y+sa,

then it becomes (ss+s~)+yS=0, if and only if m is even and s~+ss=f l sk, Eq. (24) has three distinct roots in GF(2=).

Since the codes are quasi-perfect, for any syndromes sx, ssE GF(2=), among formulas (26'), (27) and (28) there is a cri at least such that Eq. (24) has three distinct roots in GF (2m). Conversely, any trs satisfying formula (26'), formula (27) or formula (28), is substituted into Eq. (24), then Eq. (24) has three distinct roots in GF(2 ~) and the three distinct roots form an error pattern with weight 3 corresponding with syndromes s~ and ss. Thus by this method one can find all solutions of Eq. (20).

Acknowledgment I wish to acknowledge my indebtedness to Professors Cai Changnian and Zhou Jiongpan for useful discussion and helpful advice in the preparation of this paper.

References

| I ] D. C. Govenstein, W. W. Peterson and N. Zierler, Inform. Contr. 3 (1960), 291. | 2 ] F. J. MacWiUiams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, New

York, 1977, p. 279. { 3 ] C. R. P. Hartmann, IEEE Trans. on IT, IT-17 (1971), 765. [ 4 ] J. A. V. D. Horst and T. Berger, IEEE Trans. on IT, IT-22 (1976), 138. 15 ] ~ : ~ l i ~ , ~ , I - ~ [ ~ , 1976, P. 132. [ 6 ] E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968.