Skip to main content
Contents Embed Dark Mode Prev Up Next \(\require{mathtools} \setcounter{MaxMatrixCols}{15}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\GF}{\mathrm{GF}}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\wt}{wt}
\DeclareMathOperator{\im}{im}
\makeatletter
\@ifundefined{char}{
\DeclareMathOperator{\char}{char}
}{}
\makeatother
\DeclareMathOperator{\GRS}{GRS}
\DeclareMathOperator{\RS}{RS}
\DeclareMathOperator{\BCH}{BCH}
\newcommand{\transpose}[1]{#1^{\top}}
\newcommand{\by}{\mathbf{y}}
\newcommand{\bc}{\mathbf{c}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\bm}{\mathbf{m}}
\newcommand{\bs}{\mathbf{s}}
\newcommand{\be}{\mathbf{e}}
\newcommand{\bu}{\mathbf{u}}
\newcommand{\bv}{\mathbf{v}}
\newcommand{\bh}{\mathbf{h}}
\newcommand{\br}{\mathbf{r}}
\newcommand{\bzero}{\mathbf{0}}
\newcommand{\balpha}{\boldsymbol{\alpha}}
\renewcommand{\vec}[1]{\boldsymbol{#1}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\newcommand{\fillinmath}[1]{\mathchoice{\underline{\displaystyle \phantom{\ \,#1\ \,}}}{\underline{\textstyle \phantom{\ \,#1\ \,}}}{\underline{\scriptstyle \phantom{\ \,#1\ \,}}}{\underline{\scriptscriptstyle\phantom{\ \,#1\ \,}}}}
\)
Section Daily Prep 23
Today we will return to BCH codes, viewing them as cyclic codes and applying the theory we’ve learned to say more about their behavior.
Subsection Learning Objectives
Subsubsection Basic Learning Objectives
Objectives
Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.
State the definitions of designed distance, consecutive root sequence, and excess roots of a BCH code.
Determine the excess roots of a BCH code given its consecutive root sequence and use this to determine the dimension of the code.
Subsubsection Advanced Learning Objectives
Objectives
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
State and apply the BCH bound to finding a lower bound on the distance of a cyclic code.
Design BCH codes with specified parameters.
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Roth, Sections 8.4-8.5, pp. 247-256
Vanstone & van Oorschot, Sections 6.1-6.2, pp 201-210
Figure 72. Reference Video for BCH Codes as Cyclic Codes
Subsection Important Terms
