Section Week 7
This is an outline of the topics we covered in the seventh week of class.
Subsection Tuesday 2/24
Coding Theorist of the Day.
Ron Roth, DSc 1988
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At Technion in Haifa
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Broad work: MDS codes, rank metric, field structure, constrained coding
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Author of Introduction to Coding Theory
Reminders/Announcements.
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Group Contract Submission postponed to next R
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Today: Do more with operations in finite fields; introduce characteristic, order, primitive elements
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Thursday: Splitting fields and a \(2\)-error correcting code
Results on Order and Primitive Elements.
Lemma 37.
Let \(F\) be a finite field.
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For \(a \in F, a^{|F|}=a\)
Proof.
Sketch:
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If \(a=0\text{,}\) clear. If \(a\neq 0\) then\begin{equation*} \{a\alpha \mid \alpha \in F^*\} = \{\alpha \mid \alpha \in F^*\} \end{equation*}since multiplication by \(\alpha\) is invertible. So\begin{align*} \prod_{\alpha \in F^*} (a\alpha)\amp=\prod_{\alpha \in F^*} (\alpha)\\ a^{|\F^*|} \amp = 1 \end{align*}
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Follows from (1) and Degree Mantra.
Lemma 38. Order of an Element Divides \(q-1\).
Proof.
Use the Division Algorithm: \(q-1=b|a|+r\) with \(r\lt|a|\text{.}\) Then
\begin{align*}
1 \amp = a^{q-1}\\
\amp = a^{b|a|+r}\\
\amp = (a^{|a|})^b a^r\\
\amp = a^r
\end{align*}
and since \(r\lt|a|\) we must have \(r=0\text{.}\)
Theorem 39. Finite Fields have a Primitive Element.
Proof.
Sketch: Show that the elements of maximum order are primitive.
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Claim: If \(t\) is the max order of any \(\alpha \in F^*\) then for any \(\beta \in F^*\) we have \(|\beta|\mid t\text{.}\)Idea: Take \(\alpha\) with \(|\alpha|=t\text{.}\) Let \(|\beta|=s, d=\gcd(s,t)\) and write \(s=bd\) with \(\gcd(b,d)=1\text{.}\) Then
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Show \(t\) and \(b\) divide \(|\alpha \beta^d|\) by looking at \((\alpha \beta^d)^{|\alpha\beta^d|b}\) and \((\alpha \beta^d)^{|\alpha\beta^d|t}\)
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Conclude because of the GCDs that \(d=s\)
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Now count: all \(\beta\in F^*\) are roots of \(x^t-x\text{,}\) so we have \(t\geq q-1\text{.}\) But also \(t\leq q-1\) by Order of an Element Divides \(q-1\).
Subsection Thursday 2/26
Coding Theorist of the Day.
Heide Gluesing-Luerssen, PhD Bremen 1991
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At University of Kentucky
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Transitioned research to coding theory from system control theory post PhD
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Weight enumerators, cyclic codes, rank metric & subspace coding
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My thesis advisor
Reminders/Announcements.
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Office hours today only until 4
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Group Contract information posted
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Problem Set M4 posted; due 3/10. Submit by 3/5 for feedback before exam
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Exam 2 on T 3/10 over modules 3 & 4 (through today)
Goals.
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Establish the existence of all finite fields
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Give a construction for a \(2\)-error correcting code (an alternant code)
Characteristic, Frobenius, Splitting Field.
Recall the binomial theorem:
\begin{equation*}
(x+y)^p=\sum_{i=0}^p \binom{p}{i} x^i y^{p-i}
\end{equation*}
For the Frobenius map \(f_m: F \to F, \alpha \mapsto \alpha^{p^m}\) where \(\char{F}=p\text{,}\) we have that if \(|F|=p^n\) then \(f_{n-m}\) is \(f_m^{-1}\text{.}\)
Theorem 40. Fields of Order \(p^n\) Exist.
The roots of \(Q(x)=x^{p^n}-x\in\F_p[x]\) in one of its splitting fields form a field of order \(p^n\text{.}\)
Proof.
Sketch: Let \(\alpha,\beta\) be roots of \(Q(x)\) in a splitting field \(E\text{.}\)
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\begin{align*} (\alpha + \beta)^{p^n} \amp = \alpha^{p^n}+\beta^{p^n}=\alpha+\beta \\ (\alpha \beta)^{p^n} \amp = \alpha^{p^n}\beta^{p^n}=\alpha\beta \end{align*}
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Any finite subset of a field closed under field operations is a field (only requires checking inverses)
So the roots are a field of size \(p^n\text{.}\)
A 2-error correcting code.
We build off a binary Hamming code.
