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m →{{anchor|Minimum distance}}The distance d: \atop \min
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m →Lower and upper bounds of block codes: \overset{\underset{\mathrm{def}}{}}{=}
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<math>C =\{C_i\}_{i\ge1}</math> is called '' family of codes'', where <math>C_i</math> is an <math>(n_i,k_i,d_i)_q</math> code with monotonic increasing <math>n_i</math>.
'''Rate''' of family of codes
'''Relative distance''' of family of codes
To explore the relationship between <math>R(C)</math> and <math>\delta(C)</math>, a set of lower and upper bounds of block codes are known.
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For <math>q=2</math>, <math>R+2\delta\le1</math>. In other words, <math>k + 2d \le n</math>.
For the general case, the following Plotkin bounds holds for any <math>C \subseteq \mathbb{F}_q^{n} </math> with distance
# If <math>d > (1-{1 \over q})n, |C| \le {qd \over {qd -(q-1)n}} </math>
===[[Gilbert-Varshamov bound|Gilbert–Varshamov bound]]===
<math>R\ge1-H_q(\delta)-\epsilon</math>, where <math>0 \le \delta \le 1-{1\over q}, 0\le \epsilon \le 1- H_q(\delta)</math>,
<math> H_q(x) ~\
=== [[Johnson bound]] ===
Define <math>J_q(\delta) ~\
Let <math>J_q(n, d, e)</math> be the maximum number of codewords in a Hamming ball of radius
Then we have the ''Johnson Bound'' : <math>J_q(n,d,e)\le qnd</math>, if <math>{e \over n} \le {{q-1}\over q}\left( {1-\sqrt{1-{q \over{q-1}}\cdot{d \over n}}}\, \right)=J_q({d \over n})</math>
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