Week 5

Section Week 5

This is an outline of the topics we covered in the fifth week of class.

Subsection Tuesday 2/10

It was an exam. No notes today.

Subsection Thursday 2/12

From the activity: 5).

\begin{align*} |W| \amp = \sum_{i=0}^{d-2} \binom{\ell-1}{i}(q-1)^i\\ \amp = V_1(\ell-1,d-2)\\ \amp \leq V_q(n-1,d-2)\\ \amp \lt q^{n-k} \end{align*}

For 6) the distance is at least \(d\) because there is no constraint on sets of \(d\) or more columns.

🔗

Theorem 33. Fraction of [n,k,d]_q Codes.

Let \(F=GF(q)\) and \(n\geq k,d\) be positive integers. Let

\begin{equation*} \rho = \frac{q^k-1}{q-1} \frac{V_q(n,d-1)}{q^n}\text{.} \end{equation*}

Then the fraction of linear \([n,k]_q\) codes with distance strictly less than \(d\) is at most \(\rho\text{.}\)

🔗

Proof.

Sketch: Take a random \(k\times n\) matrix \(G\) and a random (uniform) \(m\in F^k\text{.}\) Then \(mG\) is a uniform random element of \(F^n\text{.}\) The probability such a vector has weight \(d-1\) or less is exactly the fraction of words of length n with distance \(d-1\) or less to the zero word. We then multiply by the number of nonzero words in the code divided by \(q-1\) since any two non-zero multiples of a codeword of weight \(d-1\) or less have the same weight.

🔗

Notes:

Usually \(\rho\) is small, so “most” linear codes meet the GV bound
🔗

🔗
For binary codes there is no known explicit construction of a code that beats the GV bound in general.
🔗

🔗
For non-binary codes, there are such constructions, called algebraic geometry (AG) codes.
🔗

🔗

🔗

Prev Top Next