19,885 research outputs found
Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex
Reed-Muller codes are among the most important classes of locally correctable
codes. Currently local decoding of Reed-Muller codes is based on decoding on
lines or quadratic curves to recover one single coordinate. To recover multiple
coordinates simultaneously, the naive way is to repeat the local decoding for
recovery of a single coordinate. This decoding algorithm might be more
expensive, i.e., require higher query complexity. In this paper, we focus on
Reed-Muller codes with usual parameter regime, namely, the total degree of
evaluation polynomials is , where is the code alphabet size
(in fact, can be as big as in our setting). By introducing a novel
variation of codex, i.e., interleaved codex (the concept of codex has been used
for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover
arbitrarily large number of coordinates of a Reed-Muller code
simultaneously at the cost of querying coordinates. It turns out that
our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that
accessing locations is in fact cheaper than repeating the procedure for
accessing a single location for times. Our estimation of success error
probability is based on error probability bound for -wise linearly
independent variables given in \cite{BR94}
Codes and Protocols for Distilling , controlled-, and Toffoli Gates
We present several different codes and protocols to distill ,
controlled-, and Toffoli (or ) gates. One construction is based on
codes that generalize the triorthogonal codes, allowing any of these gates to
be induced at the logical level by transversal . We present a randomized
construction of generalized triorthogonal codes obtaining an asymptotic
distillation efficiency . We also present a Reed-Muller
based construction of these codes which obtains a worse but performs
well at small sizes. Additionally, we present protocols based on checking the
stabilizers of magic states at the logical level by transversal gates
applied to codes; these protocols generalize the protocols of 1703.07847.
Several examples, including a Reed-Muller code for -to-Toffoli distillation,
punctured Reed-Muller codes for -gate distillation, and some of the check
based protocols, require a lower ratio of input gates to output gates than
other known protocols at the given order of error correction for the given code
size. In particular, we find a T-gate to Toffoli gate code with
distance as well as triorthogonal codes with parameters
with very low prefactors in front of
the leading order error terms in those codes.Comment: 28 pages. (v2) fixed a part of the proof on random triorthogonal
codes, added comments on Clifford circuits for Reed-Muller states (v3) minor
chang
Reed-Muller Codec Simulation Performance
The approach to error correction coding taken by modern digital communication systems
started in the late 1940’s with the ground breaking work of Shannon, Hamming and Golay. Reed-
Muller (RM) codes were an important step beyond the Hamming and Golay codes because they
allowed more flexibility in the size of the code word and the number of correctable errors per code
word. Whereas the Hamming and Golay codes were specific codes with particular values for q; n; k;
and t, the RM codes were a class of binary codes with a wide range of allowable design parameters.
Binary Reed-Muller codes are among the most prominent families of codes in coding theory. They
have been extensively studied and employed for practical applications. In this research, the
performance simulation of Reed-Muller Codec was presented. An introduction on Reed-Muller codes,
were introduced that consists of defining the key terms and operation used with the binary numbers.
Reed-Muller codes were defined and encoding matrices were discussed. The decoding process was
given and some examples were demonstrated to clarify the method. The results and the performance of
Reed-Muller encoding were presented and the messages been encoded using the defined matrices were
shown. The simulation of the decoding part also been shown. The performance of Reed-Muller codes
were then analyzed in terms of its code rate, code length and minimum Hamming distance. The
analysis that performed also successfully examines the relationship between the parameters of Reed-
Muller coding. The decoding part of the Reed-Muller codes can detect one error and correct it as
shown in the examples
Enhanced Recursive Reed-Muller Erasure Decoding
Recent work have shown that Reed-Muller (RM) codes achieve the erasure
channel capacity. However, this performance is obtained with maximum-likelihood
decoding which can be costly for practical applications. In this paper, we
propose an encoding/decoding scheme for Reed-Muller codes on the packet erasure
channel based on Plotkin construction. We present several improvements over the
generic decoding. They allow, for a light cost, to compete with
maximum-likelihood decoding performance, especially on high-rate codes, while
significantly outperforming it in terms of speed
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