Zyablov bound
In coding theory, the Zyablov bound is a lower bound on the rate r {\displaystyle r} and relative distance δ {\displaystyle \delta } that are achievable by concatenated codes. == Statement of the bound == The bound states that there exists a family of q {\displaystyle q} -ary (concatenated, linear) codes with rate r {\displaystyle r} and relative distance δ {\displaystyle \delta } whenever r ⩽ max 0 ⩽ r ′ ⩽ 1 − H q ( δ ) r ′ ⋅ ( 1 − δ H q − 1 ( 1 − r ′ ) ) {\displaystyle r\leqslant \max \limits _{0\leqslant r'\leqslant 1-H_{q}(\delta )}r'\cdot \left(1-{\delta \over {H_{q}^{-1}(1-r')}}\right)} , where H q {\displaystyle H_{q}} is the q {\displaystyle q} -ary entropy function H q ( x ) = x log q ( q − 1 ) − x log q ( x ) − ( 1 − x ) log q ( 1 − x ) {\displaystyle H_{q}(x)=x\log _{q}(q-1)-x\log _{q}(x)-(1-x)\log _{q}(1-x)} .