Some remarks on multiplicity codes

Multiplicity codes are algebraic error-correcting codes generalizing classical polynomial evaluation codes, and are based on evaluating polynomials and their derivatives. This small augmentation confers upon them better local decoding, list-decoding and local list-decoding algorithms than their classical counterparts. We survey what is known about these codes, present some variations and improvements, and finally list some interesting open problems.

[1]  Michael Alekhnovich.  Linear diophantine equations over polynomials and soft decoding of Reed-Solomon codes , 2005, IEEE Trans. Inf. Theory.

[2]  Chaoping Xing,et al.  Nonlinear codes from algebraic curves improving the Tsfasman-Vladut-Zink bound , 2003, IEEE Transactions on Information Theory.

[3]  Shubhangi Saraf,et al.  High-rate codes with sublinear-time decoding , 2011, STOC '11.

[4]  Volker Strassen,et al.  The computational complexity of continued fractions , 1981, SYMSAC '81.

[5]  Rafail Ostrovsky,et al.  Local Correctability of Expander Codes , 2013, ICALP.

[6]  Venkatesan Guruswami,et al.  Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy , 2005, IEEE Transactions on Information Theory.

[7]  Swastik Kopparty,et al.  List-Decoding Multiplicity Codes , 2012, Theory Comput..

[8]  Alexander Vardy,et al.  Correcting errors beyond the Guruswami-Sudan radius in polynomial time , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[9]  Yuval Ishai,et al.  Information-Theoretic Private Information Retrieval: A Unied Construction (Extended Abstract) , 2001 .

[10]  Madhu Sudan,et al.  Extensions to the Method of Multiplicities, with Applications to Kakeya Sets and Mergers , 2013, SIAM J. Comput..

[11]  Venkatesan Guruswami,et al.  Optimal Rate List Decoding via Derivative Codes , 2011, APPROX-RANDOM.

[12]  Xin-Wen Wu,et al.  List decoding of q-ary Reed-Muller codes , 2004, IEEE Transactions on Information Theory.

[13]  Larry Guth,et al.  Algebraic methods in discrete analogs of the Kakeya problem , 2008, 0812.1043.

[14]  Haim Kaplan,et al.  On Lines and Joints , 2009, Discret. Comput. Geom..

[15]  Francis Y. L. Chin.  A Generalized Asymptotic Upper Bound on Fast Polynomial Evaluation and Interpolation , 1976, SIAM J. Comput..

[16]  Yuval Ishai,et al.  Information-Theoretic Private Information Retrieval: A Unified Construction , 2001, ICALP.

[17]  Alan Guo,et al.  New affine-invariant codes from lifting , 2012, ITCS '13.