Low-Power Cooling Codes with Efficient Encoding and Decoding

Abstract

A class of low-power cooling (LPC) codes, to control simultaneously both the peak temperature and the average power consumption of interconnects, was introduced recently. An (n, t, w)-LPC code is a coding scheme over n wires that (A) avoids state transitions on the t hottest wires (cooling), and (B) limits the number of transitions to w in each transmission (low-power).A few constructions for large LPC codes that have efficient encoding and decoding schemes, are given. In particular, when w is fixed, we construct LPC codes of size (n/w) w−1 and show that these LPC codes can be modified to correct errors efficiently. We further present a construction for large LPC codes based on a mapping from cooling codes to LPC codes. The efficiency of the encoding/decoding for the constructed LPC codes depends on the efficiency of the decoding/encoding for the related cooling codes and the ones for the mapping.(n, t, w)-LPC code is a coding scheme for communication over a bus consisting of n wires, if the scheme has the following two features:(A) every transmission does not cause state transitions on the t hottest wires;(B) the number of state transitions on all the wires is at most w in every transmission. LPC codes have both features, while cooling codes control only the peak temperature.Definition 1. For n and t, an (n, t)-cooling code C of size M is defined as a collection {C 1 , C 2 , . . . , C M }, where C 1 , C 2 , . . . , C M are disjoint subsets of {0, 1} n satisfying the following property: for any set S ⊆ [n] of size |S| = t and for i ∈ [M], there exists a vector u ∈ C i such that supp(u) ∩ S = ∅. We refer to C 1 , C 2 , . . . , C M as codesets and the vectors in them as codewords.Using partial spreads, Chee et al. [3] constructed LPC codes with efficient encoding and decoding schemes.When t ≤ 0.687n and w ≥ (n − t)/2, these codes achieve optimal asymptotic rates. However, when w is small, i.e. low-power coding is used, the code rates are small and Chee et al. proposed another construction based on decomposition of the complete hypergraph into perfect matchings. While the construction results in LPC codes of large size, usually efficient encoding and decoding algorithms are not known.In this work, we focus on this regime (w small) and construct LPC codes with efficient encoding and decoding schemes. Specifically, our contributions are as follows.(I) We propose a method that takes a linear erasure code as input and constructs an LPC code. Using this method, we then construct a family of LPC codes of size (n/w) w−1 which attains the asymptotic upper bound O(n w−1 ) when w is fixed. We also use this method to construct a class of LPC codes of size (n/w) w−e−1 which is able to correct e transmission errors.(II) We propose efficient encoding/decoding schemes for the LPC codes of the given construction. In particular, for the above family of LPC codes, we demonstrate encoding with O(n) multiplications over F q and decoding with O(w 3 ) multiplications over F q , where q = n/w. Furthermore, the related class of LPC codes i...

Domination Mappings into the Hamming Ball: Existence, Constructions, and Algorithms

“…, y n ) such that y j = 0 for all j ∈ S. The cooling codes of [2] provide an elegant solution to this problem using spreads or partial spreads [4,12]. Domination mappings were introduced in [3] in order to simultaneously achieve both of these desirable properties. Given a subset S of [n], one can follow the edges of the domination graph G to find a subset S of [m] so that every position in S is dominated by some position in S .…”

“…If ϕ is a G-dominating mapping from {0, 1} m into B(n, w), the transmitted word y = ϕ(x) satisfies wt(y) w and y j = 0 for all j ∈ S. Since ϕ is injective, x can be recovered from y at the receiver. For more details on this encoding/decoding procedure, we refer the reader to [2,3].…”

“…In the context of coding for high-performance interconnects, the parameters m and w would be usually given, and one would like to use an (m, n, w)-domination mapping with the lowest possible n, in order to minimize the redundancy n − m of the encoding procedure. We note that although domination mappings were defined in [3], they were not studied therein. In particular, it is not clear when such mappings exist.…”

“…It is easy to verify by inspection that ϕ is G-dominating, where G = [3] ∪ [4], E) can be chosen as any one of the following three domination graphs: (2)…”

“…Note that the requirement of no isolated right vertices in Definition 1 means that every position j ∈ [n] in the image of ϕ is dominated by some position i ∈ [m] of its domain. This is motivated by the application to thermal-management coding for high-performance interconnects [2,3], briefly described in what follows.…”