An introduction to robust codes over finite fields

Shlomo Engelberg, Osnat Keren

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We consider the problem of securing data using linear and nonlinear codes over the binary numbers. We start by developing a conservation law for codes. Then we explain why linear codes, which are easy to understand and implement, are useful when protecting data from rarely occurring random errors. By a simple argument, we demonstrate that linear codes are not a good way to secure data against an attacker. Having ruled out linear codes for this purpose, we take up nonlinear codes. We explain what a finite field is and how data can be represented by elements of a finite field. We then consider codes that are nonlinear functions of the data-the elements of the finite field. We show that binary quadratic codes suffer from the same deficiencies as linear codes. Next we consider cubic codes. First, we show that cubic codes do a good job of detecting changes made by an attacker. Then we demonstrate that certain cubic codes provide a large measure of protection against attackers and some protection against certain relatively common random errors. We show that cubic codes are also reasonably efficient in a well-defined sense. Finally, we briefly consider other interesting nonlinear codes.

Original languageEnglish
Pages (from-to)751-763
Number of pages13
JournalSIAM Review
Issue number4
StatePublished - 2013


  • Finite fields
  • Nonlinear codes
  • Robust codes


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