Two new bounds on the size of binary codes with a minimum distance of three

1995 ◽  
Vol 6 (3) ◽  
pp. 219-227 ◽  
Author(s):  
Yaron Klein ◽  
Simon Litsyn ◽  
Alexander Vardy
Keyword(s):  
Minimum Distance ◽  
Binary Codes

scholarly journals A new upper bound for binary codes with minimum distance four

1998 ◽  
Vol 187 (1-3) ◽  
pp. 291-295
Author(s):  
Jun Kyo Kim ◽  
Sang Geun Hahn
Keyword(s):  
Upper Bound ◽  
Minimum Distance ◽  
Binary Codes

scholarly journals Some Remarks on the Plotkin Bound

10.37236/1746 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Jörn Quistorff
Keyword(s):  
Upper Bound ◽  
Coding Theory ◽  
Minimum Distance ◽  
Binary Codes ◽  
Plotkin Bound ◽  
Hamming Metric ◽  
Lee Metric

In coding theory, Plotkin's upper bound on the maximal cadinality of a code with minimum distance at least $d$ is well known. He presented it for binary codes where Hamming and Lee metric coincide. After a brief discussion of the generalization to $q$-ary codes preserved with the Hamming metric, the application of the Plotkin bound to $q$-ary codes preserved with the Lee metric due to Wyner and Graham is improved.

scholarly journals Improving the Gilbert-Varshamov bound for $q$-ary codes

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Van H. Vu ◽  
Lei Wu
Keyword(s):  
Recent Work ◽  
Positive Constant ◽  
Minimum Distance ◽  
Maximum Size ◽  
Binary Codes ◽  
International Audience ◽  
Positive Integers

International audience Given positive integers $q$, $n$ and $d$, denote by $A_q(n,d)$ the maximum size of a $q$-ary code of length $n$ and minimum distance $d$. The famous Gilbert-Varshamov bound asserts that $A_q(n,d+1) \geq q^n / V_q(n,d)$, where $V_q(n,d)=\sum_{i=0}^d \binom{n}{i}(q-1)^i$ is the volume of a $q$-ary sphere of radius $d$. Extending a recent work of Jiang and Vardy on binary codes, we show that for any positive constant $\alpha$ less than $(q-1)/q$ there is a positive constant $c$ such that for $d \leq \alpha n, A_q(n,d+1) \geq c \frac{q^n}{ V_q(n,d)}n$. This confirms a conjecture by Jiang and Vardy.

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