# Introduction to Coding Theory Lecture Notes

@inproceedings{Lindell2009IntroductionTC, title={Introduction to Coding Theory Lecture Notes}, author={Yehuda Lindell}, year={2009} }

These are lecture notes for an advanced undergraduate (and beginning graduate) course in Coding Theory in the Computer Science Department at Bar-Ilan University. These notes contain the technical material covered but do not include much of the motivation and discussion that is given in the lectures. It is therefore not intended for self study, and is not a replacement for what we cover in class. This is a first draft of the notes and they may therefore contain errors. These lecture notes are…

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## 44 Citations

PSPACE using Error Correcting Codes

- Computer Science
- 2010

This work shows that the IP theorem can be proved by using general error correcting codes, and believes that this establishes a rigorous basis for the intuition that underlies the use of polynomials.

Combinatorial PCPs with Short Proofs

- Computer Science, Mathematics2012 IEEE 27th Conference on Computational Complexity
- 2012

This work provides a combinatorial construction of PCPs of length n-4·lognO(loglogn), coming very close to the state-of-the-art algebraic constructions (whose proof length is n-logn O(1)), and develops a few generic PCP techniques which may be of independent interest.

List Decoding Barnes-Wall Lattices

- Computer Science2012 IEEE 27th Conference on Computational Complexity
- 2012

The results imply a polynomial-time listdecoding algorithm for any error radius bounded away from the minimum distance, thus beating a typical barrier for natural error-correcting codes posed by the Johnson radius.

Combinatorial Construction of Locally Testable Codes

- Computer ScienceSIAM J. Comput.
- 2007

This work presents a new and arguably simpler construction of LTCs that is purely combinatorial, does not rely on PCP machinery, and matches the parameters of the best known construction.

Combinatorial construction of locally testable codes

- Computer Science, MathematicsSTOC
- 2008

A new and arguably simpler construction of LTCs is presented that is purely combinatorial, does not rely on PCP machinery and matches the parameters of the best known construction, however, unlike the latter construction, the construction is not entirely explicit.

Decoding Random Linear Codes in Õ(20.054n)

- Computer Science
- 2012

A new algorithm for decoding linear codes is presented that is inspired by a representation technique due to Howgrave-Graham and Joux in the context of subset sum algorithms that offers a rigorous complexity analysis for random linear codes and brings the time complexity down to O 20.05363n.

IP = PSPACE Using Error-Correcting Codes

- Computer ScienceSIAM J. Comput.
- 2013

This work shows that the $\mathbf{IP}$ theorem can be proved by using general error-correcting codes and their tensor products and establishes a rigorous basis for the intuition that underlies the use of polynomials.

Linear-time encodable codes meeting the gilbert-varshamov bound and their cryptographic applications

- Computer ScienceITCS
- 2014

This work presents a randomized construction of linear error-correcting codes which can be encoded in linear time and yet enjoy several useful features of random linear codes, and demonstrates the usefulness of these new codes by presenting several applications in coding theory and cryptography.

Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2016

One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de-Morgan formulas. Karchmer, Raz, and Wigderson [20] suggested to approach this problem…

Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

- Computer Science, Mathematics2013 IEEE 54th Annual Symposium on Foundations of Computer Science
- 2013

This work shows that if the query complexity is relaxed to polynomial, then one can construct PCPs of linear length for circuit-SAT, and PCP of length O(tlog t) for any language in NTIME(t), and is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity.

## References

SHOWING 1-10 OF 40 REFERENCES

Introduction to the theory of error-correcting codes

- Computer ScienceProceedings of the IEEE
- 1984

Characterization of (2(q + 1) + 2, 2, t, q)-min-hyper in PG(t, q) (t 3, q 5) and its applications to error-correcting codes and the theory of rank metric is transposed.

A survey of perfect codes

- Computer ScienceAdv. Math. Commun.
- 2008

This survey paper is devoted to the rather many recent results, that have appeared during the last ten years, on perfect 1-error correcting binary codes.

A theorem on the distribution of weights in a systematic code

- Computer Science
- 1963

The spectrum of a systematic code determines uniquely the spectrum of its dual code (the orthogonal vector space) and the two sets of integers are related by a system of linear equations.

Error detecting and error correcting codes

- Computer Science
- 1950

The author was led to the study given in this paper from a consideration of large scale computing machines in which a large number of operations must be performed without a single error in the end…

A Mathematical Theory of Communication

- Computer Science
- 2006

It is proved that the authors can get some positive data rate that has the same small error probability and also there is an upper bound of the data rate, which means they cannot achieve the data rates with any encoding scheme that has small enough error probability over the upper bound.

Coding Theorems for a Discrete Source With a Fidelity CriterionInstitute of Radio Engineers, International Convention Record, vol. 7, 1959.

- Computer Science
- 1993

For a wide class of distortion measures and discrete sources of information there exists a functionR(d) (depending on the particular distortion measure and source) which measures the equivalent rateR of the source (in bits per letter produced) whendis the allowed distortion level.

On the Nonexistence of Perfect 2- and 3-Hamming-Error-Correcting Codes of GF(q)

- Mathematics, Computer ScienceInf. Control.
- 1970