# Graph Theoretic Methods in Coding Theory

Chapter

## Abstract

Let $${\Sigma }_{q} =\{ 0,1,\ldots,q - 1\}$$ be an alphabet of order q. A q-ary (unrestricted) code C of length n and size | C | is a subset of Σ q n containing | C | elements called codewords. The Hamming weight wt(c) of a codeword c is the number of its nonzero entries. A constant-weight code is a code where all the codewords have the same Hamming weight. The Hamming distance d(c, c′) between two codewords c and c′ is the number of positions where they have different entries. The minimum Hamming distance of a code C is the largest integer Δ such that ∀c,c′C, d(c, c′) ≥ Δ.

### References

1. 1.
Macwilliams FJ, Sloane NJA (1977) The theory of error-correcting-codes. North-Holland, Amsterdam
2. 2.
Sloane NJA (1989) Unsolved problems in graph theory arising from the study of codes. Graph Theory Notes of New York 18:11-20Google Scholar
3. 3.
Ahlswede R, Khachatrian LH (1997) The complete intersection theorem for systems of finite sets. Eur J Combinator 18:125–136
4. 4.
Ahlswede R, Khachatrian LH (1998) The diametric theorem in Hamming spaces - optimal anticodes. Adv Appl Math 20:429–449
5. 5.
Frankl P, Tokushige N (1999) The Erdős-Ko-Rado theorem for integer sequences. Combinatorica 19:55–63
6. 6.
Diestel R (2006) Graph theory. Springer, New YorkGoogle Scholar
7. 7.
Godsil C, Royle G (2001) Algebraic graph theory. Springer, New York
8. 8.
van Lint JH, Wilson RM (2001) A course in combinatorics. Cambridge University Press, United Kingdom
9. 9.
Ahlswede R (2001) On perfect codes and related concepts. Designs Codes Cryptography 22:221–237
10. 10.
Kleitman DJ (1966) On a combinatorial conjecture of Erdős. J Combin Theor 1:209–214
11. 11.
El Rouayheb S, Georghiades CN, Soljanin E, Sprintson A (2007) Bounds on codes based on graph theory. Int Symp Inform Theor NiceGoogle Scholar
12. 12.
Matsumoto R, Kurosawa K, Itoh T, Konno T, Uyematsu T (2006) Primal-dual distance bounds of linear codes with applications to cryptography. IEEE Trans Inform Theor 52:4251–4256
13. 13.
Tolhuizen L (1997) The generalized Gilbert–Varshamov bound is implied by turàn’s theorem. IEEE Trans Inform Theor 43:1605–1606
14. 14.
Delsarte P (1973) An algebraic approach to association schemes of coding theory. Phillips J Res 10Google Scholar
15. 15.
Agrell E, Vardy A, Zeger K (2000) Upper bounds for constant-weight codes. IEEE Trans Inform Theor 46:2373–2395
16. 16.
Johnson SM (1962) A new upper bound for error-correcting codes. IRE Trans Inform Theor IT-8:203–207Google Scholar
17. 17.
Levenshtein VI (1974) Upper bound estimates for fixed-weight codes. Probl Inform Trans 7:281–287Google Scholar
18. 18.
Erdős P, Ko C, Rado R (1961) Intersection theorems for systems of finite sets. Quart J Math Oxford 12:313–320
19. 19.
Wilson RM (1984) The exact bound on the Erdos-ko-Rado Theorem. Combinatorica 4:247–260
20. 20.
Jiang T, Vardy A (2000) Asymptotic improvement of the Gilbert–Varshamov bound on the size of binary codes. IEEE Trans Inform Theor 50(8):1655–2395
21. 21.
Richardson T, Urbanke R (2008) Modern coding theory. Cambridge University Press, Cambridge