Description
\([n,1,n]\) binary linear code encoding one bit of information into an \(n\)-bit string. The length \(n\) needs to be an odd number, since the receiver will pick the majority to recover the information. The idea is to increase the code distance by repeating the logical information several times. It is a \((n,1)\)-Hamming code. Its automorphism group is \(S_n\).Protection
Detects errors on up to \(\frac{n-1}{2}\) coordinates, corrects erasure errors on up to \(\frac{n-1}{2}\) coordinates. The generator matrix is \(G=\left[\begin{smallmatrix}1 & 1&\cdots& 1 & 1 \end{smallmatrix}\right]\).Rate
Code rate is \(\frac{1}{n}\), code distance is \(n\).Decoding
Calculate the Hamming weight \(d_H\) of an error word. If \(d_H\leq \frac{n-1}{2}\), decode the code as 0. If \(d_H\geq \frac{n+1}{2}\), decode the code as 1.Local automaton decoder for the repetition code on a 2D lattice based on Toom's rule [1–4].Local automaton decoder for the repetition code on a 1D lattice by Gacs that is translation-invariant, that does not require synchronization of local clocks, and that has a constant encoding rate [5,6].Local automaton decoder for the repetition code on a 1D lattice by Tsirelson [7].Local automaton decoder obtained from reinforcement learning [8].Fault Tolerance
Triple modular redundancy (TMR) error-correction protocol [9] for fault-tolerant memory operations and classical gate operations; see section 2.6 and 2.7 Ref. [10] for a pedagogical explanation.Threshold
Suppose each bit has probability \(p\) of being received correctly, independent for each bit. The probability that a repetition code is received correctly is \(\sum_{k=0}^{(n-1)/2}\frac{n!}{k!(n-k)!}p^{n-k}(1-p)^{k}\). If \(\frac{1}{2}\leq p\), then one can always increase the probability of success by increasing the number of physical bits \(n\); see section 2.2.1 Ref. [10] for a pedagogical explanation.The first threshold theorem for classical circuits was proven by von Neumann [11] using cellular automata [12] and which spurred the study of noisy classical circuits [13–16].Realizations
Repetition codes, in conjunction with other codes, were used in magnetic disks [17].Communication protocols such as FlexRay [18].'Cousins
- \([2^m-1,m,2^{m-1}]\) simplex code— The simplex code for \(m=2\) reduces to a four-bit repetition code.
- Perfect binary code— Repetition codes are perfect for odd \(n\).
- Quantum repetition code— A quantum repetition code can be thought of as a classical \([n,1,d]\) repetition code with classical distance \(d=\left\lfloor (n-1)/2\right\rfloor\) embedded in a quantum system.
- \([2^r-1,2^r-r-1,3]\) Hamming code— The triple repetition code \([3,1,3]\) is the smallest Hamming code.
- \(D_4\) hyper-diamond lattice— The four-bit repetition code yields the \(D_4\) hyper-diamond lattice via Construction A.
- \(E_8\) Gosset lattice— The \([8,1,8]\) repetition code can be used to obtain the \(E_8\) Gosset lattice [19; Exam. 10.7.1].
- Single parity-check (SPC) code— Binary SPCs and repetition codes are dual to each other.
- \(q\)-ary repetition code
- Compactified \(\mathbb{R}\) gauge theory code— The compactified \(\mathbb{R}\) gauge theory code is constructed from a hypergraph product of two repetition codes over the integers.
- Tiger surface code— The tiger surface code is constructed from a hypergraph product of two repetition codes over the integers.
- Self-correcting quantum code— The repetition code associated with the 2D classical Ising model is a self-correcting classical memory [20][21; Sec. V.A].
- \([[2m,2m-2,2]]\) error-detecting code— The \([[2m,2m-2,2]]\) error-detecting code is constructed via the CSS construction from an SPC code and its dual repetition code [22; Sec. III].
- Chamon model code— The Chamon model code can be obtained from a hypergraph product of three repetition codes [23], but done in a different way than the 3D surface code; see [24; Sec. 3.4].
- Sierpinsky fractal spin-liquid (SFSL) code— The Sierpinsky fractal spin-liquid code is a hypergraph product of the repetition code and the Newman-Moore code [25,26].
- Two-foliated fracton code— The two-foliated fracton code is a hypergraph product of the repetition code and the plaquette Ising code on a square lattice with periodic boundary conditions [27].
- Long-range enhanced surface code (LRESC)— LRESCs are constructed constructed using a hypergraph product of two copies of a concatenated LDPC-repetition seed code.
- Toric code— The toric code can be obtained from a hypergraph product of two repetition codes [28; Exam. 6].
- 3D surface code— The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes [29; Exam. A.1], but done in a different way than the Chamon code; see [24; Sec. 3.4].
- XZZX surface code— The Chamon model code can be obtained from a particular hypergraph product of three repetition codes [23]; see [24; Sec. 3.4]. Using only two repetition codes yields the XZZX code, making that code a 2D analogue of the Chamon code [24; Sec. 2].
Member of code lists
Primary Hierarchy
Parents
The repetition code is cyclic with generator polynomial \(1+x+\cdots+x^{n-1}\).
RM\((0,m)\) are repetition codes.
Repetition code
References
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Page edit log
- Bao Bach (2023-02-22) — most recent
- Victor V. Albert (2022-09-28)
- Victor V. Albert (2022-05-27)
- Victor V. Albert (2021-12-15)
- Yijia Xu (2021-12-14)
Cite as:
“Repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/repetition