Partition code (Section 4.1)
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(r,ℓ)
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\(\frac {r}{r + \ell ^{2}}\)
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\(n - k + 1 - \ell \left (\frac {k\ell }{r} - 1\right)\)
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Product code (Section 4.2)
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(r,ℓ)
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\(\left (\frac {r}{r+1}\right)^{\ell }\)
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ℓ+1
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Concatenated code (Section 5)
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(r,ℓ)
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\(\frac {\ell + 2}{\ell + 4}\frac {r}{r + 2}\)
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ℓ+1
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Regular bipartite graph-based code (Section 6.1)
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(r,ℓ)
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\( \geq \frac {r - \ell }{r + \ell }\)
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≥ℓ+1
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Unbalanced bipartite expander graph-based code (Section 6.2.1)
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(r,ℓ)
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\(\geq 1 + \frac {h}{\Delta }\frac {r}{\ell } - h\)
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\(\geq \left (2 - \epsilon -\frac {\epsilon }{t}\right)\alpha n\)
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Double cover of regular expander graph based-code (Section 6.2.2)
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(r,ℓ)
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\(\geq 2\frac {r}{\ell \Delta } - 1\)
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\(\delta \left (\delta - \frac {\lambda }{\Delta }\right)n\)
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Hadamard code (Section 7)
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(r=ℓ+1,ℓ), \(\forall 1\le \ell \le \frac {n-1}2\)
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\(\frac {\log (n+1)}{n}\)
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\(\frac {n+1}2\)
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