Results are presented in the form of 14 theorems specifying sufficient conditions under which it is possible to construct new more efficient single- and multi error correcting codes from existing one when the method used to construct existing codes is known

Varshamov, R. R.

English

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' . L 
NASA TECHNICAL MEMOruurnUM NASA TM-75204 
POSSIBILITIES FOR IMPROVING THE EFFICIENCY 
OF LINEAR ERROR-CORRECTING CODES 
R. R. Varshamov 
(NASA-TM-75204) fOSSIBILI~IES FOR IMPROVING N78-17746 
BE EFFICJENCY OF lINEAR ~ERC£ -CORRECTING 
CODES (Naticna1 Aeronautics ard Space 
Administraticn) 9 p CSCl 098 Unclas 
00/63 05470 
Translation of "0 vozmozhnostyakh uvelicheniya moshchnosti 
lineynykh korrektiruyshchikh kodov", Doklady 
Akademii Nauk SSR, vol. 223, 0.1, 
1975, pp. 60-63. 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
WASHINGTON, D.C. 20546 NOVEMBER 1977 
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https://ntrs.nasa.gov/search.jsp?R=19780009803 2020-03-22T06:22:24+00:00Z
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1 
POSSIBILITIES FOR IMPROVING THE EFFICIENCY 
OF LINEAR ERROR-CORRECTING CODES 
R. R. Varshamov 
Computing Center of the Acad e my of Sci e nc e s of the Arm e nia n SSR 
and Yerivan State University 
The elaboration of design methods for the construction of 
noiseproof coding systems with a maximal data transmission rate 
(or equivalently with maximum efficiency of code subsets) when 
the main parameters of the system are specified is one of the 
main tasks of algebraic coding theory. Here n is the length of 
a code block, t is the number of detected errors, and q is the 
base of the code. This note presents some results which are rele-
vant to possibilities for constructing, within Hamming distance, 
noiseproof systems of signals in which the efficiency of code s ub s e t s 
exceeds many times the efficiency of all similar known linear codes 
described in the literature. 
The following theorem states our basic concept. 
Theorem 1. Regardless of the magnitude of the number e 
and method used to construct error-correcting codes correcting 
t symmetric errors, values of the parameters nand q can always 
be found (on the basis of this method) for which the following is 
true: 
a ) new classes of codes can be constructed in which the efficiency 
of code subsets is more than e times greater than the efficiency 
of the corresponding codes that were constructed using the original 
method. 
b ) the error-correcting capacity of the old codes is preserved. 
we will assume that a method for constructing coding systems 
correcting a fixed number t of errors is given if a completely de-
fined method which allows the construction of such systems is 
specified for any arbitrary nand q. 
* Numbers in margin indicate pagination in the f oreign text 
The physical interpretation of Theorem 1 will be discussed 
using Hamming and Bose-Choudhuri-Hocquenghem codes as examples. 
1. Hamming Codes. It is well known [1] tha t Hamm.ing codes 
represent the class o f linear codes correcting single symmetric 
errors. Although these codes arc optimal in the class o f linear 
codes, they belong to a class of codes for which a construction 
method has been developed. Therefore, according to the statement 
of theorem 1, new classes of codes allowing a n improvement in the 
efficiency of Hamming codes (even without limit) can be constructed. 
Henceforth we will denote by g(n,q) the efficiency of the op-
timal Hamming code of length n with base q. We will also use the 
notat i on E (n) = q [logqn] where [x] is the greatest integer not q 
greater than x, and {x} = x - [x]. 
The case q = 31 
Theorem 2. For any integers n, satisfying the inequalities 
signal systems with base 3 correcting single symmetric errors exist 
in which the efficiency of code subsets b(n,3) is strictly greater 
than the e f ficiency of the corresponding Hamming codes and satis-
fies the inequality 
b (n , 3) >3l-(luK, .n }g (n, 3) . / 61 
This result can be stated in somewhat stronger form, namel y a s 
the following theorem, 
Theorem 3. Values of the parameter n can always be found for 
which coding systems with base 3 correcti n g s in g l e err ors can be con-
s tructed in whi chthe e ffi c i e ncy of code s u b s ets b (n, J ) satisfies t h e 
i n equali ty 
where o (n) = 1 or 2. 
Thus, for example, the following theorem holds: 
1 Our method is inef ficient in the case q = 2 
2 
1 
--~~ 
--~ - -'---~- -----~----:'--
Theorem 4. 
the form ;? k + 5 
Let m = Pl ••• p , P , s = 1, ••• , 0 , be primes of 
(J s 
and gs be a primitive root modulo Ps' 
1) , -I m - ln , Ill , (J , Ill, .11, = - __ 
t1 
p, - l 
Jl =~ , { I O~ I III} > lo r 3:2, 
'1 
Then for any integers a and n (E: 3 (m) < 2n< m/2), the set of 
all possible solutions of the congruence 
t' m, 1 p' 
\1 \ '\ n 1 (n " ' I' /,) /' 
..,,-J .....-.- __ J ."-, I • 'I , + ~(1l10d 111), 
) \. ~ I 
2 
were x , un, taking on 
u 
arbitrarily the values 0, 1 and 2, ,is a 
code of length n with 
Among these codes the 
base 3, correcting single symmetric errors. 
efficiency of the optimal code is 
form 
b( ") - .,, Il, " -' II (' ''V "' )"( 11 " ) 3 (') , tI 1 6(1l) I. 
The following theorem is also valid. 
Theorem 5. Let m = PI ••• Pa 
g k + 3, m > 3 (2m) and q = 3. 
, where p are primes of the 
s 
Then for arbitrary integers a , B and n ( E: 3(2n) < 2n < m), the 
set of all possible solutions of the system of congruences 
where x = 0, 1 
u 
o ,. - t :! J " 
\ ) 1 \ ( J I k ~'+Jll' ) ,r _1I + . .. _ -:;( (Ill(l(l m), 
w ,,--J 
~ I lu I 
n 
~ xu= p (mod2 ) , 
u - ! 
and 2 (u ~n) and Xu = 0 (u >n), is a code of 
length n correcting single symmetric errors. Among these codes, 
the efficiency of the optimal code is 
},( II,:n>;~ I -'l ": £:(11 . :1), l( l1 ) .) 
We will now consider the general case of arbitrary q v 3 = p. 
'2 x = 0 for all u > n 
u 
~ p is an odd prime 
3 
~ 
I 
- ~- --------- .., 
Theorem 6 
inequalities 
( v> l). For any integer values n, satisfying the 
1(J~; (q - 1) -I q __ {In~ I} '-:: I I ", 
signal systems with base q correcting single symmetric errors 
exist in which the efficiency of code subsets b(n,q) satisfies the 
relation 
b(n, q) - p '- I:!(n, j) 
Theorem 7 (v = 1). For any integer values n satisfying the ine-
qualities 
signal systems with base p exist in which the efficiency of code 
subsets b(n,p) satisfies the inequality 
b(n, p»p l - I " '~ p ""'C"llg(n, p) 
This result can be stated in somewhat stronger form. 
/ 62 
Theorem 8. Values of n exist such that codes with base p 
correcting single symmetric errors can be constructed in which the 
efficiency of code subsets satisfies the inequality 
2. Bose-Chandhuri-Hocquenghem (B. C. H.) codes. B. C. H. codes 
are a generalization of Hamming codes to the multi-error correcting 
case, and like Hamming codes, they belong to a class of codes for 
which a construction method exists. Therefore, according to our 
concept, new more efficient codes correcting multiple errors can 
be constructed on the basis of these codes. 
Let us denote by Gt(n,q) the efficiency of optimal B. C. H. 
codes of length n with base q correcting t symmetric errors. 
The case q = 2, t = 2. 
4 
\ 
1-
5 
Theorem 9. For any integers n ({log2n} < log2(2/(3» and 
q = 2, codes correcting binary symmetric errors can be constructed 
whose efficiency B2 (n,2) is strictly greater than the efficiency 
of the corresponding B. C. H. codes and satisfies the inequality 
11, (n, 2) >~ 1_ (1 0 ,. "/jj G2 (n, 2). (1) 
Thus, for example, the following theorem holds. 
Theorem 10. Let p be a prime of the form 6 k-l satisfying 
the inequality {log2 p} < log2 (2/ Ij) . 
Then for any integers a JS , Y u and n (£ 2(P) ~ n5 p), the set 
of all possible solutions of the system of congruences 
n 
,E ux"".. a (modp ), 
U I 
n I>2Xu sa ~ (modp), 
" ~ :I:" ES "( (morl3), 
where x = 0 or 1, is a code of length n with base 2 correcting 
u 
two symmetric errors. Among these codes, the efficiency of the 
optimal codes is strictly greater than the efficiency of analogous 
B. C. H. codes, and it satisfies inequality 1. 
The case q = 3, t = 2. 
Theorem 11. Let n be any integer satisfying the inequality 
{log3n) ~ log33/2 and g be a primitive element of the Galois field 
GF (£ 3 ( 3n) ) . 
Then the set of all possible solutions of the equations 
" I: g"x lI= b, 
,,=1 u _.t 
where Xu ¢ GF ( 3) and a, b ¢ GF ( £ 3 ( 3n) ), is a code of length n with 
J 
1 
( . . 
base 3, correcting two symmetric errors, whose efficiency B2 (n,3) 1S 
B z (n , 3) = ( 1'3)1_(-1) (log, nJ G
2 
(n, 3). 
The case q = 4, t = 2. 
Theorem 12. Suppose an arbitrary natural number n > 5 and a 
primitive element g of the Galois field GF(c 4 (12n», are given. 
Then the set of all possible solutions of the e qua tion s 
" 
"g"x =b ~.,.J " , 
4 
where Xu ¢ GF( x ) and a, b,~ GF(E4 (12n» is a code of length n with 
base 4 correcting two symmetric errors, whose efficiency strictly 
exceeds the efficiency of the analogous B. C. H. code and satisfies 
the relation 
However, on the basis of only the Bose-Chaudhuri-Hocquenghem 
results, one can also prove the following theorem. 
Case of arbitrary p. 
Theorem 13. For any arbitrary values t ~ 2, p ~ 2t and n 
( t lOgpn) < logp(p/v (n», l~ v (n) < 2), codes of length n with 
base p, correcting t symmetric errors can be constructed, whose 
efficiency Bt(n,p) is strictly greater than the efficiency of the 
corresponding B. C. H. codes correcting the same number of errors, 
and whose efficiency satisfies the i nequa lity 
Note that when n approaches infinit y, the quantity v(n) takes 
on the value 1 infinitely ma ny times. Hence the following theorem 
holds. 
Theorem 14. For fixed t and p ~ 2t, the inequality 
B,(n, p»(pl-( Io,p nl ) 2I-IG,(n, p) 
is satisfied infinitely many times as n approaches infinity. 
-- ~ ---- ----
6 
• • •• \00 -
. 
Peterson, W. 
codes) . 
REFERENCES 
Kody, ispra v1yayushchiye oshibki (Error-correcting 
Mir Press, Moscow, 1964. 
COPYRIGHT : Izdate1 ' stvo Nauka, ])OKLADY AKADEMII NAUK SSSR 1977 
7 
STANDARD TITLE PAGE 
1. Report No. 2. Government Access I on No . 3. ReCIpient', Catalog No . 
NASA TM-7S204 
4. Title and Subtitle 5. Report Dote 
POSSIBILITIES FOR IMPROVING THE Novembp.r lQ77 
EFFICIEN CY OF LINEAR ERROR-CORRECTING 6 . Performing Organ ization Code 
CODES 
-;;-
7 . Author( .) 8. Performing Organ i za ti on Report No. 
R. R. Varshamov 
10. Work Unit No . 
11. Contract or Grant No . 
9. Performino Orgoni zotion Nome and Address NASw-2790 
Leo Kanner Associates 13. Type 0 f Report and Per i od Covered 
Redwood City, California 94063 Translation 
12. Spon.oring Agency Name and Address 
National Aeronautics and Space Adminis 14 . Sponsoring Agency Code 
tration, Washington, D.C. 20546 
15. Supplement"ry Note. 
Translation of "0 vozmozhnostyakh uvelicheniya moshchnosti 
lineynykh korrektiruyushchikh kodov, " Doklady Akademii Nauk 
SSR, vol. 223, o. 1, 1975, pp. 60-63. 
16. Abstract 
Results are presented in the form of 14 theorems specifying 
sufficient conditions under which it is possible to con-
struct new more efficient single- and multi-error correct-
ing codes from existing ones when the method used to con-
struct .existing codes is known. 
17. Key Word. (Selected by AutIlOr{s») 18. Distribution Sta l emen t This copyrighted 
Soviet work is reproduced and sold by 
NTIS unde r license from VAAP, the 
Soviet copyright agency . No further 
copying is permitted without 
pe r mission from VAAP . 
19 . Security Classif. (of this report) 20 . Se cur i ty C loss if . (of Ih i . page) 21. No. of Pages 22. Pri ce 
Unclassified Unclassified 9 


Possibilities for improving the efficiency of linear error-correcting codes

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