Lectures

Following is an outline of lectures given along with references and links to additional reading. For the abbreviations (ECT), (Hill), (Roth), (Lint), please check the references page.

\( \def\PAL{\rm PAL} \def\HP{\rm HP} \def\MP{\rm MP} \def\EQ{\rm EQ} \def\USELESS{\rm USELESS} \def\EMPTY{\rm EMPTY} \def\DTIME{\sf DTIME} \def\NTIME{\sf NTIME} \def\DSPACE{\sf DSPACE} \def\NSPACE{\sf NSPACE} \def\P{\sf P} \def\co{\sf co} \def\NP{\sf NP} \def\LOG{\sf LOG} \def\NLOG{\sf NLOG} \def\PSPACE{\sf PSPACE} \def\NPSPACE{\sf NPSPACE} \def\EXP{\sf EXP} \def\F{\mathbb{F}} \)

Lec 1 (30 Jul, Thu) : Course plan. Basic goals in the design of error correcting codes. Shannon’s model of communication. Examples of simple codes - repetition codes, parity check codes. On error detection and correction of codes.

^{Reading : (ECT) Section 1.1; (Hill) Chapter 1}

^{Person of the day : Claude E. Shannon - Father of information theory, AI agent of same first name, Mathematical Theory of Communication, Creative thinking.}

Lec 2 (01 Aug, Sat) : (9.30 - 10.45 AM) Hamming code- description and codewords. Parity check criteria for error detection. Error correction in Hamming codes as guided exercise.

^{Reading : (ECT) Section 1.2 to 1.5}

^{Person of the day : Richard W. Hamming - Famous for Hamming codes, Hamming metric, Hamming ball and Hamming bound, ACM Turing award laudation ‘Purpose of computing is insight, not numbers’, You and your research.}

^{Exercises : (ECT) Chap 1, Problems 1.1, 1.3}

Lec 3 (01 Aug, Sat) : (11.00 - 12.30 PM) Hat check puzzle - two approaches using codes. Minimum distance and its role in error detection and unique decoding.

^{Reading : (ECT) Section 1.2}

^{Exercises : (ECT) Chap 1, Problems 1.5, 1.10, 1.11}

Lec 4 (04 Aug, Tue) : Code with distance $d \ge 2$ $\iff$, it can detect $d-1$ errors, correct $\lfloor (d-1)/2 \rfloor$ errors. Errors and erasures. BSC and BEC channel models.

^{Reading : (ECT) Chap Section 1.2, 1.3, 1.4, (Hill) Chap 2}

^{Person of the day : Robert Fano - Famous for Fano’s inequality, Developed one of the first timer-sharing computers. One of the world’s first open-source advocates.}

^{Exercises : (ECT) Chap 1, Problem 1.6, 1.8}

Lec 5 (06 Aug, Thu) : Linear binary codes. Review of basic linear algebra. Linear codes. Characterization using generator and parity check matrix. Computing minimum distance from parity check matrix.

^{Reading : (ECT) Chap 2, Section 2.2, 2.3, 2.4, (Hill) Chap 5}

^{Person of the day : Irving S. Reed - Famous for Reed Solomon codes, Design team of MADDIDA computer, Developed the register transfer language.}

^{Exercises : (Hill) Chap 5, Ex. 5.1, 5.3}

Lec () (11 Aug, Tue) : Instructor out of town (compensated on Aug 01)

Lec () (13 Aug, Thu) : Instructor out of town (compensated on Aug 01)

Lec () (18 Aug, Tue) : Instructor out of town (to be compensated)

Lec () (20 Aug, Thu) : Instructor out of town (to be compensated)

Lec 6 (25 Aug, Tue) : Recap. Equivalences of linear codes and equivalence in terms on operations on generator matrix. Systematic code and systematic generator matrix. Relation to parity check matrix. Decoding - MLD algorithm.

^{Reading : (Hill) Chap 5, (ECT) Chap 2, Section 2.3, 2.4, 2.5}

^{Person of the day : Gustave Solomon - Famous for Reed Solomon codes, Founder of algebraic theory of error correcting codes.}

^{Exercises : (ECT) Ex. 1.8}

Lec 7 (27 Aug, Thu) : Characterization of distance of code using linear (in)dependence of columns of parity check matrix. Decoding algorithm for linear codes. Syndrome decoding. Hardness of linear code decoding problem (statement only). Generalized Hamming codes and an efficient decoding algorithm. Limits on the relation between rate and distance - the Hamming bound. Hamming bound is tight for Generalized Hamming codes. Perfect codes.

^{Reading : (Hill) Chap 5, (ECT) Chap 2, Section 2.3, 2.4, 2.5}

^{Person of the day : Marcel J. E. Golay - Discovered binary and ternary Golay codes - perfect codes. Invented Golay Detector used to detect aircraft by emitting IR. Introduced complementary sequences which is used in Wifi and 3G standards}

^{Exercises : (ECT) Ex. 2.7, 2.10, 2.14}

Lec 8 (01 Sep, Tue) : Linear algebra fails over $\mathbb{Z}_m$ for composite $m$ - failure of existence of inverse of elements. Way to overcome this, a motivating example - constructing $\mathbb{C}$ from $\mathbb{R}$. Constructing fields that are finite. The three views of this construction - generator, polynomial, vector space views. Irreducible polynomials. Enumeration of irreducible polynomials over $\mathbb{F}_2$ of degree $2$, $3$ and $4$. Using them to construct finite fields of size 4 and 8.

^{Reading : (Roth) Chapter 3. (Hill) Chapter 3,4}

^{Person of the day : Robert J. McEliece - Showed that the decoding problem is NP-hard. Known for McEliece’s Theorem. Also McEliece cryptosystem based on hardness of decoding}

^{Exercises : (Hill) Ex. 4.1, 4.3, 4.4}

Lec () (20 Oct, Tue) : Holiday due to Dussehra