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7. Reed-Muller codes
Synopsis. The minimum distance of a perfect code cannot exceed 7 unless the code is a
repetition code. This is disappointingly low. In this nal section of the course, we construct
Reed-Muller codes, a famil
MATH32031 Coding Theory
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Synopsis. We conclude the section on cyclic codes by proving the theorem about generator and check matrices for a cyclic code. We dene two Golay codes and give a complete
classication of perfect codes up to parameter equivalenc
MATH32031 Coding Theory
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Synopsis. We dene the Reed-Muller codes R(r, m) as certain subspaces of the Boolean
algebra on V m . We calculate the parameters of R(r, m).
Week 11, lecture 1
Denition (Reed-Muller code)
The rth order Reed-Muller code on V m ,
MATH32031 Coding Theory
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Synopsis. We prove an important theorem about the properties of a dual code C . We
introduce a new decoding algorithm for linear codes called syndrome decoding. It requires
the knowledge of a check matrix of the code.
Week 07, l
MATH32031 Coding Theory
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5. Hamming codes
Synopsis. Hamming codes are essentially the rst non-trivial family of codes that we shall
meet. We start by proving the Distance Theorem for linear codes we will need it to determine the minimum distance of a Ha
MATH32031 Coding Theory
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Synopsis. By calculating the parameters of a Ham(s, q) code explicitly, we prove that Hamming codes are a family of perfect codes with minimum distance 3. We show that syndrome
decoding works for Hamming codes in an especially s
MATH32031 Coding Theory
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Synopsis. We would like to study cyclic codes using an algebraic structure which is richer
than just a vector space. An appropriate structure is a ring. But the ring Fq [x] of polynomials
is too big (innite). New, smaller rings
MATH32031 Coding Theory
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Synopsis. We prove that every cyclic code C is an ideal generated by a unique monic polynomial called the generator polynomial. We also dene a check polynomial of C. We can
classify cyclic codes of length n by listing all monic
MATH32031 Coding Theory
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4. Dual codes
Synopsis. We dene the dual code C of a linear code C. If C has a generator matrix in
standard form, we learn how to nd a generator matrix of C .
Week 07, lecture 1
Denition (inner product)
For u, v Fn , the scalar