In today’s lecture, we defined the code composition method of *code concatenation*. By picking the outer code to be on the Singleton bound and the inner code to be on the GV bound, we obtained the Zyablov bound. That is, for relative distance , we can obtain a rate of at least

where . In the class today I presented the formula with . However, the form above is more accurate since the Varshamov bound achieves a rate of for any .

Further, we presented a polynomial time construction of codes that achieves the Zyablov bound: the outer code is the strongly explicit RS code while the inner code on the GV bound can be constructed in (singly) exponential time. (The latter result was problem 4 in the homework.)

The material covered in today’s lecture can be found in the scribed notes of Lecture 24 from Fall 07.

Next lecture, we will look at a strongly explicit construction of an asymptotically good binary code due to Justesen.

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