Dual of (evaluation) codes
summary
Gives an explicit algebraic recipe: the dual of an evaluation code uses the same evaluation points, complementary monomials coming from the footprint (compute a Gröbner basis) and multiplicative scalars equal to the indicator polynomial leading coefficients. Applies to Reed‑Solomon, Reed‑Muller, Cartesian and tensor‑product families, recovers multivariate Goppa descriptions and yields concrete repair/decoding constructions (trace-based reductions of downloaded data and CCS‑T pair constructions via square and dual intersections).
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chapters (14)
- 0:00Evaluation codes and duals
- 5:45What is an evaluation code
- 8:18All codes as evaluation codes
- 11:54Minimum distance bounds
- 13:44CCS-T codes importance
- 15:52Duals via ideals
- 19:27Reed-Solomon and Reed-Muller duals
- 26:21Hermitian and norm-trace duals
- 28:17McEliece and Goppa codes
- 30:48Multivariate Goppa and tensor products
- 33:04Improving Reed-Muller codes
- 37:03Repair schemes and Cartesian codes
- 37:45Q&A
- 40:03Summary
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