Partition code (Section 4.1)

(r,ℓ)

\(\frac {r}{r + \ell ^{2}}\)

\(n  k + 1  \ell \left (\frac {k\ell }{r}  1\right)\)

Product code (Section 4.2)

(r,ℓ)

\(\left (\frac {r}{r+1}\right)^{\ell }\)

ℓ+1

Concatenated code (Section 5)

(r,ℓ)

\(\frac {\ell + 2}{\ell + 4}\frac {r}{r + 2}\)

ℓ+1

Regular bipartite graphbased code (Section 6.1)

(r,ℓ)

\( \geq \frac {r  \ell }{r + \ell }\)

≥ℓ+1

Unbalanced bipartite expander graphbased code (Section 6.2.1)

(r,ℓ)

\(\geq 1 + \frac {h}{\Delta }\frac {r}{\ell }  h\)

\(\geq \left (2  \epsilon \frac {\epsilon }{t}\right)\alpha n\)

Double cover of regular expander graph basedcode (Section 6.2.2)

(r,ℓ)

\(\geq 2\frac {r}{\ell \Delta }  1\)

\(\delta \left (\delta  \frac {\lambda }{\Delta }\right)n\)

Hadamard code (Section 7)

(r=ℓ+1,ℓ), \(\forall 1\le \ell \le \frac {n1}2\)

\(\frac {\log (n+1)}{n}\)

\(\frac {n+1}2\)
