## Description

LDPC code whose parity check matrix is constructed explicitly (i.e., non-randomly) from a particular graph [1,2] or an algebraic structure such as a combinatorial design [3–5], balanced incomplete block design [6], a partial geometry [7], or a generalized polygon [8,9]. The extra structure and/or symmetry [10] of these codes can often be used to gain a better understanding of their properties.

## Parent

## Children

## Cousins

- Combinatorial design code — Combinatorial designs can be used to construct explicit LDPC codes [3–5].
- Protograph LDPC code — Some deterministic protograph LDPC codes [11] can be obtained from the theory of voltage graphs [12,13].

## References

- [1]
- G. A. Margulis, “Explicit constructions of graphs without short cycles and low density codes”, Combinatorica 2, 71 (1982) DOI
- [2]
- C. A. Kelley, D. Sridhara, and J. Rosenthal, “Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights”, IEEE Transactions on Information Theory 53, 1460 (2007) DOI
- [3]
- S. J. Johnson and S. R. Weller, “Regular low-density parity-check codes from combinatorial designs”, Proceedings 2001 IEEE Information Theory Workshop (Cat. No.01EX494) DOI
- [4]
- S. J. Johnson and S. R. Weller, “Construction of low-density parity-check codes from Kirkman triple systems”, GLOBECOM’01. IEEE Global Telecommunications Conference (Cat. No.01CH37270) DOI
- [5]
- S. J. Johnson and S. R. Weller, “Resolvable 2-designs for regular low-density parity-check codes”, IEEE Transactions on Communications 51, 1413 (2003) DOI
- [6]
- B. Vasic and O. Milenkovic, “Combinatorial Constructions of Low-Density Parity-Check Codes for Iterative Decoding”, IEEE Transactions on Information Theory 50, 1156 (2004) DOI
- [7]
- S. J. Johnson and S. R. Weller, “Codes for iterative decoding from partial geometries”, Proceedings IEEE International Symposium on Information Theory, DOI
- [8]
- P. O. Vontobel and R. M. Tanner, “Construction of codes based on finite generalized quadrangles for iterative decoding”, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252) DOI
- [9]
- Z. Liu and D. A. Pados, “LDPC Codes From Generalized Polygons”, IEEE Transactions on Information Theory 51, 3890 (2005) DOI
- [10]
- Tanner, R. Michael, Deepak Sridhara, and Tom Fuja. "A class of group-structured LDPC codes." Proc. ISTA. 2001.
- [11]
- C. A. Kelley, “On codes designed via algebraic lifts of graphs”, 2008 46th Annual Allerton Conference on Communication, Control, and Computing (2008) DOI
- [12]
- C. A. Kelley and J. L. Walker, “LDPC codes from voltage graphs”, 2008 IEEE International Symposium on Information Theory (2008) DOI
- [13]
- L. W. Beineke et al., editors , Topics in Topological Graph Theory (Cambridge University Press, 2009) DOI

## Page edit log

- Victor V. Albert (2023-05-09) — most recent

## Cite as:

“Algebraic LDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/algebraic_ldpc