By Xuan Guang, Zhen Zhang (auth.)
There are major ways within the conception of community blunders correction coding. during this SpringerBrief, the authors summarize probably the most very important contributions following the vintage method, which represents messages by way of sequences just like algebraic coding, and likewise in brief speak about the most effects following the opposite technique, that makes use of the idea of rank metric codes for community mistakes correction of representing messages through subspaces. This booklet begins by way of setting up the elemental linear community blunders correction (LNEC) version after which characterizes similar descriptions. Distances and weights are outlined with the intention to represent the discrepancy of those vectors and to degree the seriousness of error. just like classical error-correcting codes, the authors additionally practice the minimal distance interpreting precept to LNEC codes at each one sink node, yet use specific distances. For this deciphering precept, it truly is proven that the minimal distance of a LNEC code at every one sink node can absolutely signify its error-detecting, error-correcting and erasure-error-correcting features with recognize to the sink node. additionally, a few vital and important coding bounds in classical coding thought are generalized to linear community errors correction coding, together with the Hamming certain, the Gilbert-Varshamov certain and the Singleton sure. numerous confident algorithms of LNEC codes are offered, really for LNEC MDS codes, besides an research in their functionality. Random linear community blunders correction coding is possible for noncoherent networks with error. Its functionality is investigated through estimating top bounds on a few failure percentages via examining the data transmission and mistake correction. ultimately, the elemental conception of subspace codes is brought together with the encoding and deciphering precept in addition to the channel version, the boundaries on subspace codes, code development and deciphering algorithms.
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Example text
ZΔ (x, yt )) achieves the minimum amongst all error vectors z ∈ Z satisfying z · Gt = z(x, yt ) · Gt . Thus, Eq. 14) can further be written as (t) Dd (yt ) = arg min w(t) (z(x, yt )) x∈X = arg min wH (zH (x, yt )) x∈X (t) (t) (t) (t) = arg min wr (zr (x, yt )) x∈X = arg min wΔ (zΔ (x, yt )). 17) are called minimum weight decoding (or minimum network Hamming weight decoding), minimum Hamming weight decoding, minimum rank decoding, and minimum dimension decoding at the sink node t ∈ T , and denoted by (t) (t) (t) D(t) (·), DH (·), Dr (·), and DΔ (·), respectively.
For a regular linear network error correction code, let a channel set {e1 , e2 , · · · , eCt } be a minimum cut between the source node s and the sink node t with an upstream-to-downstream order (ancestral order) e1 ≺ e2 ≺ · · · ≺ eCt , where Ct is the minimum cut capacity between s and t. Further let an error pattern be ρ = {eω , eω +1 , · · · , eCt }, where again ω is the information rate. Then Φ (t) ∩ Δ (t, ρ ) = {0}. Proof. Let x and z represent a source message vector and an error vector, respectively.
For these distance functions, we further have the following result. 1. The distance functions dH (·, ·), dr (·, ·), dΔ (·, ·), and dw (·, ·) are equivalent, that is, for any two vectors y1 , y2 ∈ Yt = F |In(t)| , (t) (t) (t) (t) dH (y1 , y2 ) = dr (y1 , y2 ) = dΔ (y1 , y2 ) = dw (y1 , y2 ). Proof. Let y1 , y2 ∈ Yt be arbitrary two received vectors at the sink node t ∈ T . , (t) (t) wH (z) ≥ wr (z) ≥ wΔ (z) ≥ w(t) (z), where again ρz is the error pattern induced by the error vector z. This further implies (t) (t) (t) (t) dH (y1 , y2 ) ≥ dr (y1 , y2 ) ≥ dΔ (y1 , y2 ) ≥ dw (y1 , y2 ).