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 9 (03 Sep, Thu) : Description of distance $5$ code using $GF(16)$ and a decoding algorithm. Basics of algebra - well ordering principle, division algorithm, GCD and its properties for integers. polynomial division and polynomial GCD computation.

^{Reading : (Roth) Chapter 3, Class notes}

^{Person of the day : Selmer M. Johnson - Famous for Johnson bound, Johnson graphs, Johnson schemes.}

^{Exercises : Work out more questions from the sage demo done in class.}

Lec 10 (08 Sep, Tue) : Polynomial division and polynomial GCD algorithm. Irreducibility and divisibility of polynomials. Reviewed proof of unique factorization of integers (fundamental theorem of arithmetic). Argued polynomials in $\mathbb{F}[x]$ are uniquely factorizable.

^{Reading : (Roth) Chapter 3, Class notes}

^{Person of the day : Elwyn R. Berlekamp - Noted algebraic coding theorist and entrepreneur. Famous for algorithms to factor polynomials over finite fields and integers, known for two algorithms for decoding Reed-Solomon codes.}

Lec 11 (10 Sep, Thu) : Basics of groups, cosets. Properties of cosets. Used them to derive Lagrange’s theorem. Extension fields and why study them. Towards proving cyclic nature of finite groups.

^{Reading : (Roth) Chapter 3, Class notes}

^{Person of the day : Neil A. Sloane - Creator of an enormous encyclopedia for interesting integer sequences oeis.org launched in 1996. Has contributions in combinatorics, coding theory and sphere packing.}

Lec 12 (15 Sep, Tue) : Roots and multiplicity. Proved that any non-zero polynomial $p$ in $\mathbb{F}[x]$ has at most $deg(p)$ many roots in any extension of $\mathbb{F}$. Order of an element in a group and three of its properties. Towards arguing that $\mathbb{F}^*$ is cyclic using the results seen so far.

^{Reading : (Roth) Chapter 3, Class notes}

^{Person of the day : Michael Tanner - Founder of the field of “codes on graphs”. Creator of Tanner code. These codes are used in cellular, wifi and deep-space communication systems.}

Lec 13 (17 Sep, Thu) : Recap of discussion so far. Relating order of an element to the order of its power. Order of product of two elements whose orders are relativly prime is exactly the product of the orders. Used these results to complete the proof that $\mathbb{F}^*$ is cyclic.

^{Reading : (Roth) Chapter 3, Class notes}

^{Person of the day : Jessie C. MacWilliams - She is known for the MacWilliams identity. Author of the earliest comprehensive text “The Theory of Error-Correcting Codes” with Neil A. Sloane. This book was of interest for mathematicians as well as engineers. First woman to publish in the area of coding theory.}

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