188 IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 35, NO. 1, JANUARY 1989

where T(G):=card{L: d i m L = k - 1 and c a r d G n L = i } . (If d‘= k + 1, i.e., if any k columns of 9 are linearly inde- pendent, we have simply uiqI = ( :,)( ; I ) ( - l )k- l - ‘ . )

all L of dim L I k -2, i.e., U,(’)= 0 whenever j I k -2.

To compute First Case: k - 1 I s < n. Here we clearly have Is ( L ) = 0 for

It is well-known that

we consider two cases.

if i = s otherwise. m = s ‘ ( A)( T ) ( - l ) n l - ’ = (i:

This yields ul? I = T~ (G) and, via (4‘),

Some Convolutional Codes Whose Free Distances are Maximal

KHALED A. S. ABDEL-GHAFFAR

Abstract-The free distance of a convolutional code of rate 1 / 7 7 is bounded by n times the constraint length of its encoder. Two classes of convolutional codes whose free distances meet this bound are studied. A technique of constructing convolutional codes that meet this bound for all sufficiently large n is also given.

I. INTRODUCTION

A,, -, = card P, = ( 4 - 1) . T ~ ( C ) .

Second Case: k - 1 > s. It is well-known that

(7) Let go(x),...,g,,-l(x) be polynomials over GF(q). Then G ( x ) = ( g , ( x ) ; . ., g,,- l(x)) is an encoder of a q-ary convolu- tional codes V of rate l / n . This convolutional code is defined by

and from (3) and (7) it follows that m

= a , x i , a, E GF( q ) , r is an integer), r = r I

r = k - l f ?(‘).( k L 1 ) = ( k l l ) ’ (9)

Substituting (8) into (6 ) and eliminating Th , (G) with the help of (9) yields

and its elements are called codewords. A series n ( x ) = I ~ = r a , x i , where r is an integer, is called a Laurent series. The Hamming weight of the Laurent series a ( x ) is the number of its nonzero coefficients (could be infinite) and is denoted by w ( a ( x ) ) . If a,( x), . . . , a,, (x) are Laurent series, we define

2, = ( - 1) A - w( %(X) 1 . . ., % I ( X ) ) = w( d x ) ) + . ’ . + W( a , , - , ( x > ) . - 1

n - 1 The free distance of V is

d, = min { W( C( x)) : C( x) is a nonzero codeword}. r = k

An encoder C( x) such that gcd ( go( x), . . . , g,, ~ ( x ) ) = 1 is said to be noncatastrophic. To find the free distance of a code generated by a noncatastrophic encoder G ( x ) , it suffices to consider codewords a ( x ) C ( x ) where a ( x ) is a polynomial, i.e.,

d, = min { w( a( x) G( x)) : a( x) is a nonzero polynomial) .

- ( - 1 ) h - \ - I k -1 ( s ) ( k : l ) -

n - 1

(‘‘1 +( 1 T , ( G ) . ( f,( 2;;;’) r = k

where the final simplification is achieved using the identities The constraint length Of c ( x ) is defined to be’

( , i - s j - ( f - s - l j = ( i ; S ; l ) i - k + l ~ - - k + l

Substituting (6) and (10) into (4‘) proves our assertion for the remaining cases s < k - 1.

Remark: If d‘= k + 1, taking into account the aforementioned simplification of .A?,, our proof gives a straightforward deriva- tion of the weight enumerator of MDS codes by the principle of inclusion and exclusion. A derivation of the weight enumerator of MDS codes by the principle of inclusion and exclusion that involves an additional Mobius inversion formula appears in [2, theorem 4.2.81.

L = l t max degg,(x) 0 5 r s n - 1

Since W( C( x ) ) = E::(; W( g, (n)) is at most equal to nL, we have d! 5 nL. In this correspondence, we are interested in codes for whch d, = nL. In this case, all the coefficients of the polynomi- als g , ( x ) ; . ., g,,-l(x) are nonzero.

Of course, C(x) = (1,. . .,l) generates a q-ary code of free distance n. If q = 2 and L 2 2, then there is no code with free distance nL. Viterbi [SI, [9] constructed codes of free distances nL where L = 2 and q > 2. Justesen [3] proved that if d, = nL, then L I q. He also constructed codes with d, = nL for n I q - 1 and L / q I F( n) for some function F( n) such that 1/3 I F( n) - < (5 - 6 ) / 4 = 0.7. Other constructions are given in [2], [SI.

We give two new classes of codes with d, = nL. The first construction is described in Section 11. The construction works

REFERENCES Manuscript received June 9, 1987: revised April 4, 1988. This work was

supported in part by a grant from the pacific ~ ~ l ~ ~ i ~ Foundation. The author was with the IRM Research Division, Almaden Research Center,

San Jose. CA. He is now with the DeDartment of Electrical Eneineerine and

[1]

[2]

H. R. Halder and w. Heise, Einfuhrung in die Kurnhinutorik. Miinchen: Hanser, 1976 (also Berlin: Akadernie. 1977). J . H. van Lint, Codittg T h e o v , Lecture Notes in Mathematics. no. 201. Berlin: Springer, 1971.

[ 3 ] F. J. MacWilliams and N. J. A. Sloane, The Theorv of Error-Correcting Codes.

[41 D . w. Newhart, “On minimum weight codewords in QR codes.” J . Comhrn. Theory, Ser. A, vol. 48. pp. 104-119, 1988.

Computer Science. College of Engineering, University of California, Davis, CA 95616.

Amsterdam, The Netherlands: North Holland, 1977. ;EEE Log Number 8825356. Many authors, including those of [3] and (51, define the constraint length to

be t i l . . The definition stated here agrees with that of [7].

0018-9448/89/0100-0188$01.00 01989 IEEE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 35, NO, 1, JANUARY 1989 189

for 1 < L I q whenever n is sufficiently large. The second con- struction described in Section I11 gives codes with L _< n + 1 I q.

11. CONSTRUCTION A If G ( x ) = ( g o ( x ) , . . . , g,, -, ( x ) ) is a generator of a convolu-

tional code of rate l / n and free distance nL, where L is the constraint length of G( x ) , then go( x ) , . . . , g;- ( x ) are polyno- mials of degree L - 1 with nonzero coefficients. It is interesting to consider the case in which g , (x ) ; . ., g n - l ( x ) are all the polynomials satisfying this property. In such a case n = ( q - l)L. In this section, we prove the following theorem.

Theorem 1: Let g 0 ( x ) ; . ., g , , - l ( x ) be all the polynomials over GF(q) of degree L - 1 with nonzero coefficients where n =

( q - l ) ' . , q f 2 , and L I q . Let C,,,(x)=(go(x),...,g,,-,(x)). Then G,, / . ( x ) generates a convolutional code of rate l / n and free distance nL. Moreover, a ( x ) is a Laurent series such that W ( a ( x ) G , , , ( x ) ) = n L if and onlyif

where m is an integer, a, E GF(q) - {0), and u1 E GF(q).

To prove Theorem 1 , we start with some lemmas. The follow- ing lemma appears in [l] in a slightly different form where a construction of block codes similar to the construction of convo- lutional codes described here is given.

where t is a positive integer, be nonzero elements in GF( q). Define

Lemma I : Let a,; . . , a,

N ( t ) = c W(a,bo+ . . . +nl - lb , - l ) . bo,. ' .. hr-1 EGF(q)-(O)

Then,

ProoJ The proof is by induction on t. The lemma holds for t = 1. Assume that it holds for general t. If aobo + . . . + a,- lb;- = 0, which occurs ( q - 1)' - N ( t ) times as bo,. . . , br- run over the nonzero values of GF(q), then W(aobo+ . . . + ~ , _ , b ~ - ~ + a,b,) = 1 for any b, # 0. On the other hand, if aobo + . . . + a , ~ , b , _ , f O (which occurs N ( t ) times as b , ; . . , b , - , run over the nonzero values of GF(q)), then W(aobo + . . . + a, - b, + a, h, ) = 0 for a unique nonzero b,. Hence

and the inductive step is proved. 0

In the following lemma, we derive an expression for the weight of a codeword a(x )G, , l . ( x ) .

Lemma 2: Let a ( x ) =E~=, ,u,x' be a power series over GF(q). For each integer i define s, = W@>=,u ,x ' ) . Let C(x) =

a(x )G, , / , ( x ) , where Cy, l - ( x ) is defined in Theorem 1. Then

m W ( C ( X ) ) = N ( s , - s , - l ~ ) ( q - l ) ~ ~ - ' , + ' , - , .

r = O

Proof: Let C( x ) = ( c,( x ) , . . . , c , ~ ( x ) ) , where c, ( x ) =

a ( x ) g , ( x ) . Write g,(x)=Cf::g, , ,x ' and c , ( ~ ) = E ~ = ~ c , , , x ' . Then

where a, = 0 for t < 0. Hence

n - 1

By the definition of the polynomials go( x ) , . . . , g,, - (x), it fol- lows that

c

n - 1

- - h,- I + ,.' . , h, t GF(q) - ( 0 )

Since [ { t : i - L + 1 I t I i, a, f 0)l =s, - s , - ~ . , Lemma 1 gives

c h , - , + l.'. . , h, t GF(q) - ( 0 )

c - - h, E GF(q)- (0): U, t 0

m

We now complete the proof of Theorem 1.

Proof of Theorem 1: Since G , , [ . ( x ) is a noncatastrophic encoder for q f 2, it suffices to consider polynomial u ( x ) to find the free distance of the code. Indeed, from Lemma 2, W( a ( x)G4. 1. (x)) is infinite if a( x ) is not a polynomial. Hence assume a ( x ) = Z : = , a , x ' , U,, a, # 0. From Lemmas 1 and 2, we get

If the weight of C ( x ) = a ( x ) G , . ( x ) equals the free distance of the code, then s, > s , - ~ for 011 I L + r - 1 . Indeed, if s, =s,-[ for some I , O I z I L + r - l , then = a , = O and thus we can write a ( x ) = a ' ( x ) + x r + l a " ( x ) , where a ' ( x ) and a " ( x ) are nonzero polynomials and the degree of a ' ( x ) is at most I - L. Hence

is the sum of the weights of two nonzero codewords. By defini- tion, this sum is larger than the free distance of the code. Thus we can assume s , - s , - ( , > O for O s i s L + r - l . If s , - s , - / 2 2 ,

190 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 35, NO. 1, JANUARY 1989

we get If W(a(x)g(x)) 2 L , then by Corollary 1, W(a(x)G(x)) 2 rnL + s L = vL . If W(a(x)g(x)) < L , then degn(x) 2 2, and by Corollary 1, W(a(x)G,l,,(x)) 2 nL + l . In this case, it follows from (2) that

S , - s , - L - l

( - 1) I ( q - 1 ) " - J = ( q - 1) - ( q - 1) J - 0

s n - s , _ , - 3 W( a( x)G( x)) 2 r( nL +1) + s 2 rnL + r

2 rnL + nL > ( r n + s ) L = vL + ( - l ) y q - l ) ' - J J = o

2 ( q - 1 ) -( q - 1) ' ~ - l .

( - 1) J ( q - 1) I , - I = ( q - 1)

If s, - s , - ~ . =1, then s, - S , - [ . -1

j = O

Since ao, a,. # 0, we have so - s - ~ = s L + r - l - s , - ~ =l. If there is an i for which s, - s, 2 2, we have r 2 1 and in this case (1) implies

L + r - 2 S , - s , - L - l

W(C(X)) =2(q-1)'.+ ( - l ) ' ( q - l )L- J

2 2( q - 1) + ( L + r -2)[ ( q - 1 ) - ( q - 1) L - l ]

2 L( q -l)L +( q - L ) ( q -1)- 2 L( q -1)"

I - 0

since q 2 L . Equality holds in this case if and only if r =1, i.e., a ( x ) = a , + a , x and L = q . If s , - ~ , - ~ = l for all O < i i L + r - 1 , then r = O and a(x)=a,. In this case, W(C(x))=

It is interesting to note that the number of codewords a( x)G,. L ( x) of weight d,, where a(x) is a power series and a(0) # 0, is q - 1 for L < q and q( q - 1) for q = L. This number is minimum among all codes with d, = nL as can be deduced from the proof of [3, Lemma 11. It is nice to have a small number of such codewords since the error probability of a code can be bounded by a bound proportional to this number [7]. The follow- ing is a corollary to Theorem 1.

(x) are all the monic polynomials over GF(q) of degree L - 1 with nonzero coefficients. In this case n = ( q -

In the following theorem and its proof, we show how to construct codes of free distance nL for all sufficiently large n.

Theorem 2: Let q # 2 be a prime power, L I q, and n =

( q - l ) ' ~ - ' . Then for every v 2 n2L, a q-ary convolutional code exists of rate l / v and free distance V L which has an encoder whose constraint length is L.

a I . - l x I . - l , where a1;. GF( q) . If a E GF( q ) - {0}, then

(1 - ax)g( x) = I + ( a1 - a ) x + a1(a2 - a ) x 2 +

L ( q - l)'.. 0

Corollary I : Theorem 1 holds if go (x), . . . , g,,

l .

Proof: Let g(x) = 1 + a l x + a la2x2 + . . . + a1 . . . are distinct nonzero elements in

+ alaz .. . aL-2(aL-1 - a)xL- ' + a1 .. . aLP1axL.

Since al , . . . , a,- are distinct, then W((1- ax)g( x)) 2 L. Hence if W(a(x)g(x)) < L for a nonzero polynomial a(x), then the degree of a ( x ) is at least two. Now suppose v 2 n 2 L . Write v = rn + s, 0 I s < n. Let go(x),. . . , g,,- 1( x) be the polynomials of Corollary 1 and define G,l, I , ( x ) = (g0(x);. ., gn-l(x)). Let

r S

since s < n and v 2 n2L imply r 2 nL. 0

111. CONSTRUCTION B The codes constructed in the previous section are of very low

rate. In this section, we show how to construct codes of rate l / n , where L I n t l I q .

where q # 2. Let L I n + 1 I q and n 2 2. Let u 0 ; . ., u L - l , U , , . . ., u , , ~ ~ benonzero elements in GF(q). Define gJ(x) = uJZ::,u,aJx', 0 I j~ n - 1 , and assume that gcd(g,(x);..,g,-,(x)) =l. Then G(x) = (go( x), . . . , g,, - (x)) generates a convolutional code of rate l /n and free distance nL.

Proof: Let n ( x ) be a power series such that a(0) # 0. Let C(x) = a(x)G(x) = (co(x);. ., C,,-~(X)) where c,(x) =

a(x)g,(x), 0 I J I n - 1 . Write u(x) = C:=oa,x', gJ(x) =

Cf=-,'gl, /XI, and cl (x) = Z:=occ,, /XI . Note that g,. = U, u, ai. Then we have

Theorem 3: Let GF(q) = (0, a0; . .,

I

~ , - ~ g , , ~ = u ~ ~ , - ~ u ~ a ; , f o r O I i I L - 1

a,-,g,,,=u, a,-,u,a;, f o r i z L .

t = O t = O ct,,- L - 1 L - 1

t = O t = O

a , - , u r x t , f o r O s z s L - 1

-I I

Hence, if we define

t = O L - 1

a ,_ ,u ,x ' , for I 2 L , I ' t = o

b,(x) =

then c,, = v,b,(a,) for i 2 0. Thus

( c,.o 3 . . . 9 c,, ,I - 1 ) = ( U 0 4 ( ao) 7 . . . 9 U,,- lb, ( a,,- 1 1) . Note that (c,,~,. . + , c , , , ~ ~ ) is a codeword in a generalized Reed-Solomon code [4]. Since C(x) is a noncatastrophic en- coder, to find the free distance of the code, let a ( x ) be a polynomial, i.e., a (x) =Z:=,a,x', a, f 0 for some nonnegative integer r . We consider two cases.

1) r 2 L: For 0 I I I L - 1 , b, ( x ) is of degree at most I . Also b, (x) # 0 since a, and the U ' S are nonzero. Hence b , ( x ) has at most I zeros. Thus we have

(3) W( c , . ~ ~ , . . . , c ~ , ~ - ~ ) 2 n - I ,

On the other hand, if r I I I r + L - 1, then

b,(x) = a , ~ , ~ , x ' = x ' - ~ U ~ - , U , - ~ + ~ X ~ = X ' ~ ~ b, ' ( x )

where b,'(x) = C ~ = + ~ - l - l a , ~ r u , - , + r x r . Since a, f 0, we have b:(x)#Oand b:(x)hasatmost r + L - 1 - r zeros. Hence

for 0 I I I L -1.

L - 1 r + L - 1 - 1

r=o r = O

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 35, NO. 1, JANUARY 1989 191

Thus w( C( x)) = w( CO( x) 7 . . ’ 7 c,,- 1 ( x))

r + L - 1

= c w(c, ,rJ,... , C , , , , - I )

2 c w(C,,m..’3c, . , , - l)

I = o L-1

1=0

r + L - 1

+ c w ~ , , 0 7 . ’ . ~ ~ , . , 2 - 1 ) . i = r

From (3) and (4), L - l r t L - 1

W ( C ( X ) ) ~ c ( n - i ) + ( n - r - L + l + i ) I = ” ~ = r

= nL +( n + 1 - L ) L 2 nL since L I n + 1.

2) r I L - 1: Using a similar argument, we obtain

w( Cl.0 1 . . . 3 c, . , I - 1 )

n - i, for 0 I i I r -1

for L I i I r + L - 1. 2 n - r , for r I i I L - 1 (5) i n - r - L + 1 + i ,

Hence w( c( x ) ) = w( CO( x ) 3 . . ., c,,- I( x))

r t L - l

= c w(c! .o , . . .>c , ,n - l ) . 1 = O

Using the bounds in (5), we get

W( C( x ) ) 2 nL +( n + I - L ) r 2 nL. 0

The following two results are simple corollaries to Theorem 3. Corollary 2: Let GF(q) be a field of characteristic p. Let

L = tp’ I n + 1 I q, n 2 2, for some integers t and e such that l s t s p - 1 , e>O.Then,

G ( x) = (( x - a()) ‘. - l , . . . ,( x - a,,.. ,) -1)

where a,, , . . . , a,, erates a convolutioal code of rate l / n and free distance nL.

are distinct nonzero elements in GF( q) , gen-

Proof: We have L - 1

( x - a , ) “ - ’ = ( L ; l ) ( - a / ) ’ , - ’ - ’ x ! . I = o

Let U, = a; ~ U , = ( ; 1. Then the corollary follows from Theo-

rem 3 if we show that (‘.;‘I $ 0 (modp) for 0 s i I L - 1. Since L=tpe, thep‘ lL-k i fandonly i fp‘ lk f o r c i e a n d l i k i L - 1. Hence,

I

0

k = l

Corollary 3: Let L I n + 1 I q, n 2 2. If L # q - 1 or L = n =

q - 1, then

1 -1 x I . - a I . - a ( , l - I ) I .

G ( x ) = ~ ~ . . . ~ a” 1 7 1 x - 1 ’ x - a i

where a is a primitive element in GF(q), generates a convolu- tional code of rate l / n and free distance nL.

Proof: We have

The corollarv follows Theorem 3 if we show that

If L # q - 1 , then gcd(x‘ -1, XI- - a‘)laL -1. Since a is primi- tive in GF( q) , we have a‘. # 1 and gcd(x‘- - 1, XI - a‘ ) = 1 and (6) holds. If L = q - 1, then

q - 2 ~- - n ( x e d ) .

x - a ‘ k = O k + J

Hence (6) holds if n = q - 1. 0

ACKNOWLEDGMENT The author would like to thank the referees for their helpful

suggestions.

[91

REFERENCES

T. Bier, “A family of nonbinary linear codes,” Discrete Muth., vol. 65, pp. 47-51, 1987. J. Justesen. “New convolutional code constructions and a class of asymp- totically good time-varying codes,” IEEE Truns. Inform. Theorv, vol. IT-19, pp. 220-225, Mar. 1973. -, “An algebraic construction of rate l / u convolutional codes,” IEEE Truns. Inform. Theory, vol. IT-21, pp. 577-580, Sept. 1975. F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-Correcting Codes. J. L. Massey, D. L. Costello, and J. Justesen, “Polynomial weights and code constructions,” IEEE Truns. Inform. Theory. vol. IT-19, pp. 101-110, Jan. 1973. J. L. Massey and M. K. Sain, “Inverses of linear sequential circuits,” IEEE Truns. Comput.. vol. C-17, pp. 330-337, Apr. 1968. R. J. McEliece. The Theorv of In/ormution and Coding. Reading. MA: Addison-Wesley, 1977. J. P. Odenwalder. “Dual-k convolutional codes for noncoherent demodu- lated channels,” in Proc. IEEE In t . Telecomniunicutions Con/. . 1976. pp. 165-176. A. J . Viterbi, “Two constructive classes of convolutional codes for multi- ple-signal channels,” presented at the IEEE Int. Symp. Inform. Theory, CA. Jan. 1972.

Amsterdam, The Netherlands: North-Holland, 1977.

Comments and Additions to “Robust Transmission of Unbounded Strings Using Fibonacci Representations”

RENATO M. CAPOCELLI, SENIOR MEMBER, IEEE

Abstract-An ambiguous point in the subject article is clarified. Fur- thermore, new codes that have better asymptotic performances and better synchronization capability are introduced.

Manuscript received July 19, 1987; revised March 14. 1988. This work was supported in part by F.O.R.M.E.Z. and in part by the Italian Ministry of Education.

The author was with the Department of Computer Science, Oregon State University. Cowallis. OR. He is now with the Dipartimento di Informatica ed Applicazioni, Universita di Salerno. Salerno. Italy 84100.

IEEE Log Number 8825708.

0018-9448/89/0100-0191$01.00 01989 IEEE