Section Results
This page contains theorems and propositions used throughout the course.
Subsection Results
Proposition 8 Relation Between Minimum Distance and Error-Correcting Capability
Claim 25 Closest Codeword in Standard Array
Claim 26 Standard Array Unique Decoding Condition
Theorem 30 Classification of Perfect Codes
Theorem 31 Singleton Bound
Theorem 33 Fraction of [n,k,d]_q Codes
Proposition 34 Properties of Irreducibles
Claim 35 Polynomials Mod an Irreducible Form a Field
Proposition 36 Degree Mantra
Theorem 39 Finite Fields have a Primitive Element
Theorem 40 Fields of Order \(p^n\) Exist
Theorem 49 Hamming or Sphere-Packing Bound
Theorem 50 Gilbert-Varshamov Bound for General Codes
Proposition 64 The Exponent of an Irreducible is the Order of its Roots
Proposition 65 Relation between Order of Root and Minimum Polynomial Degree
Theorem 77 Gilbert-Varshamov Bound for Linear Codes
Theorem 78 Polynomial Division & Congruence
Theorem 81 Bezout’s Identity
Theorem 91 Finite Fields of the Same Size are Isomorphic
Theorem 92 Fundamental Theorem of Cyclic Codes
