On Perfect Codes in the Johnson Graph
نویسندگان
چکیده
In this paper we consider the existence of nontrivial perfect codes in the Johnson graph J(n, w). We present combinatorial and number theory techniques to provide necessary conditions for existence of such codes and reduce the range of parameters in which 1-perfect and 2-perfect codes may exist.
منابع مشابه
On the Nonexistence of Perfect Codes in the Johnson Scheme
Although it was conjectured by Delsarte in 1973 that no nontrivial perfect codes exist in the Johnson scheme, only very partial results are known. In this paper we considerably reduce the range in which perfect codes in the Johnson scheme can exist; e.g., we show that there are no nontrivial perfect codes in the Johnson graph J(2w qp, w), p prime. We give theorems about the structure of perfect...
متن کاملThe (non-)existence of perfect codes in Lucas cubes
A Fibonacci string of length $n$ is a binary string $b = b_1b_2ldots b_n$ in which for every $1 leq i < n$, $b_icdot b_{i+1} = 0$. In other words, a Fibonacci string is a binary string without 11 as a substring. Similarly, a Lucas string is a Fibonacci string $b_1b_2ldots b_n$ that $b_1cdot b_n = 0$. For a natural number $ngeq1$, a Fibonacci cube of dimension $n$ is denoted by $Gamma_n$ and i...
متن کاملOn Perfect Codes and Related Concepts
The concept of diameter perfect codes, which seems to be a natural generalization of perfect codes (codes attaining the sphere–packing bound) is introduced. This was motivated by the “code–anticode” bound of Delsarte in distance regular graphs. This bound in conjunction with the recent complete solutions of diametric problems in the Hamming graphHq (n) and the Johnson graph J (n, k) gives a sha...
متن کاملPerfect Matchings in Edge-Transitive Graphs
We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in G, is the graph constructed from by deleting edges with an en...
متن کاملOn the oriented perfect path double cover conjecture
An oriented perfect path double cover (OPPDC) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each arc of $G_s$ lies in exactly one of the paths and each vertex of $G$ appears just once as a beginning and just once as an end of a path. Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete Math. 276 (2004) 287-294) conjectured that ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1004.5195 شماره
صفحات -
تاریخ انتشار 2010