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Algebraic Coding Theory MATH 4012 Spring 2026

Dr. Hunter Lehmann

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Worksheet Problem Set M4

Instructions: You may type up or handwrite your work, but it must be neat, professional, and organized and it must be saved as a PDF file and uploaded to the appropriate Gradescope assignment. Use a scanner or scanning app to convert handwritten work on paper to PDF. I encourage you to type your work using the provided template.
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All tasks below must have a complete solution that represents a good-faith attempt at being right to receive engagement credits. If your submission is complete and turned in on time, you will receive full engagement credits for the assignment. All other submissions will receive zero engagement credit. Read the guidelines at Grading Specifications carefully.
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To abide by the class’ academic honesty policy, your work should represent your own understanding in your own words. If you work with other students, you must clearly indicate who you worked with in your submission. The same is true for using tools like generative AI although I strongly discourage you from using such tools since you need to build your own understanding here to do well on exams.
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1. Computations in GF(4).

Let \(F=\GF(2)\) and let the field \(E=\GF(4)\) be represented as \(F[\alpha]/(\alpha^2+\alpha+1)\text{.}\)
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(a)

Let \(u_0,u_1,v_0,v_1 \in F\text{.}\) Show that the product of the elements \(u_0+u_1\alpha\) and \(v_0+v_1\alpha\) in \(E\) is given by
\begin{equation*} (u_0v_0+u_1v_1)+(u_0v_1+u_1v_0+u_1v_1)\alpha \end{equation*}
and so, it can be computed using three additions and four multiplications in \(F\text{.}\)
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(b)

Show that two elements in \(E\) be multiplied using four additions and three multiplications in \(F\text{.}\)
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Hint.
Consider the terms \(u_0v_0, u_1v_1\text{,}\) and \((u_0+u_1)(v_0+v_1)\text{.}\)
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(c)

Show that the square of an element in \(E\) can be computed using one addition in \(F\text{.}\)
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(d)

Show that the mapping \(E\to E\) defined by \(x\mapsto \alpha x^2\) can be computed only by repositioning of elements of \(F\text{,}\) without any arithmetic operations in \(F\text{.}\)
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2. The Vandermonde Matrix.

Let \(\beta_1, \beta_2, \ldots, \beta_r\) be \(r\) distinct elements of a field \(F\) and let \(V\) be the \(r\times r\) matrix given by
\begin{equation*} V = \begin{pmatrix} 1 \amp 1 \amp \ldots \amp 1\\ \beta_1 \amp \beta_2 \amp \ldots \amp \beta_r \\ \beta_1^2 \amp \beta_2^2 \amp \ldots \amp \beta_r^2 \\ \vdots \amp \vdots \amp \vdots \amp \vdots \\ \beta_1^{r-1} \amp \beta_2^{r-1} \amp \ldots \amp \beta_r^{r-1} \\ \end{pmatrix} \end{equation*}
In this problem you will show that
\begin{equation*} \det(V)= \prod_{1\leq i \lt j \leq r} (\beta_j - \beta_i)\text{.} \end{equation*}
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(a)

By careful choice of row operations and cofactor expansion, show that \(\det(V)=\det(A)\) with
\begin{equation*} A = \begin{pmatrix} \beta_2 - \beta_1 \amp \beta_3 - \beta_1 \amp \ldots \amp \beta_r - \beta_1 \\ \beta_2(\beta_2 - \beta_1) \amp \beta_3(\beta_3 - \beta_1) \amp \ldots \amp \beta_r(\beta_r - \beta_1) \\ \beta_2^2(\beta_2 - \beta_1) \amp \beta_3^2(\beta_3 - \beta_1) \amp \ldots \amp \beta_r^2(\beta_r - \beta_1) \\ \vdots \amp \vdots \amp \vdots \amp \vdots \\ \beta_2^{r-2}(\beta_2 - \beta_1) \amp \beta_3^{r-2}(\beta_3 - \beta_1) \amp \ldots \amp \beta_r^{r-2}(\beta_r - \beta_1) \\ \end{pmatrix} \end{equation*}
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(b)

Use induction on \(r\) and the previous part to prove the claim.
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3. Linearized Polynomials.

Let \(F= \GF(q)\) and \(E = \GF(q^n)\text{.}\) A linearized polynomial over \(E\) with respect to \(F\) is a polynomial over \(E\) of the form
\begin{equation*} a(x) = \sum_{i=0}^t a_i x^{q^i} \end{equation*}
for some \(t\geq 0\text{.}\) For the rest of this problem, assume all linearized polynomials are over \(E\) with respect to \(F\text{.}\) Sometimes we write \(x^{[i]}\) for \(x^{q^i}\) to avoid double superscripts.
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(a)

Let \(a(x)\) be a linearized polynomial. Show that the mapping \(\phi: E\to E\) defined by \(x\mapsto a(x)\) is a linear transformation over \(F\text{.}\)
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(b)

Show by a counting argument that every linear transformation \(L:E \to E\) over \(F\) can be represented as a mapping \(x\mapsto a(x)\) for some linearized polynomial with \(t \leq n-1\text{.}\)
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(c)

Let \(a(x)\neq0\) be a linearized polynomial with \(0\leq t \lt n\text{.}\) Fix a basis \(B=(\alpha_1\,\alpha_2\,\ldots\,\alpha_n)\) of \(E\) over \(F\text{,}\) and let \(A\) be an \(n\times n\) matrix representation of the mapping \(E\to E\) defined by \(x\mapsto a(x)\text{;}\) that is, if \(\bu\) is a column vector in \(F^n\) that represents an element \(u \in E\) as \(u=B\bu\text{,}\) then \(A\bu\) is a vector representation of \(a(u)\text{,}\) i.e., \(a(u)=BA\bu\text{.}\) Show that \(\rank(A)\geq n-t\text{.}\)
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Hint.
Find an upper bound on the size of the right kernel of \(A\text{.}\)
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4. Linearized Polynomials & Subspaces.

Throughout this problem suppose \(a(x)\) is a linearized polynomial over \(\GF(q^s)\) with respect to \(\GF(q)\text{.}\)
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(a)

Suppose the roots of \(a(x)\) lie in the extension field \(\GF(q^n)\) for \(n\geq s\text{.}\) Show that
  1. The roots of \(a(x)\) form a \(\GF(q)\)-subspace of \(\GF(q^n)\text{.}\)
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  2. If \(r\) divides \(n\) then the roots of \(a(x)\) which lie in \(\GF(q^r)\) form a \(\GF(q)\)-subspace of \(\GF(q^n)\text{.}\)
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(b)

Let \(\beta_1, \beta_2,\ldots, \beta_m\) be elements of \(\GF(q^s)\) which are linearly independent over \(\GF(q)\text{.}\) Show that the \(m\times m\) matrix
\begin{equation*} B = \begin{pmatrix} \beta_1 \amp \beta_2 \amp \ldots \amp \beta_m \\ \beta_1^q \amp \beta_2^q \amp \ldots \amp \beta_m^q \\ \beta_1^{q^2} \amp \beta_2^{q^2} \amp \ldots \amp \beta_m^{q^2} \\ \vdots \amp \vdots \amp \vdots \amp \vdots \\ \beta_1^{q^{m-1}} \amp \beta_2^{q^{m-1}} \amp \ldots \amp \beta_m^{q^{m-1}} \\ \end{pmatrix} \end{equation*}
is invertible by observing that \((a_0\, a_1\, \ldots\, a_{m-1})B=\bzero\) if and only if the \(\beta_i\) are roots of the linearized polynomial with coefficients given by the \(a_i\) and applying the previous problems.
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(c) Subspace Polynomials.

Let \(V\) be a \(\GF(q)\)-subspace of \(\GF(q^n)\text{.}\) Show that the polynomial
\begin{equation*} p_V(x) = \prod_{\gamma \in V} (x-\gamma) \end{equation*}
is a linearized polynomial over \(\GF(q^n)\) with respect to \(\GF(q)\text{.}\)
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Hint.
Let \(\{\beta_1,\beta_2,\ldots,\beta_m\}\) be a basis for \(V\) over \(\GF(q)\text{.}\) Then use the previous parts.
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