Improved upper bounds for partial spreads

High temperatures have dramatic negative effects on interconnect performance and, hence, numerous techniques have been proposed to reduce the power consumption of on-chip buses. However, existing methods fall short of fully addressing the thermal challenges posed by high-performance interconnects. In this paper, we introduce new efficient coding schemes that make it possible to directly control the peak temperature of a bus by effectively cooling its hottest wires. This is achieved by avoiding state transitions on the hottest wires for as long as necessary until their temperature drops off. We also reduce the average power consumption by making sure that the total number of state transitions on all the wires is below a prescribed threshold. We show how each of these two features can be coded for separately or, alternatively, how both can be achieved at the same time. In addition, error-correction for the transmitted information can be provided while controlling the peak temperature and/or the average power consumption.In general, our cooling codes use n > k wires to encode a given k-bit bus. One of our goals herein is to determine the minimum possible number of wires n needed to encode k bits while satisfying any combination of the three desired properties. We provide full theoretical analysis in each case. In particular, we show that n = k + t + 1 suffices to cool the t hottest wires, and this is the best possible. Moreover, although the proposed coding schemes make use of sophisticated tools from combinatorics, discrete geometry, linear algebra, and coding theory, the resulting encoders and decoders are fully practical. They do not require significant computational overhead and can be implemented without sacrificing a large circuit area.

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“…Proof: Since > MRRW(δ) by (33) and β(δ) MRRW(δ) by (32), it follows that > β(δ) and hence Inequality (22) is satisfied.…”

Section: A Asymptotic Behavior Of Cooling Codesmentioning

confidence: 96%

“…The value of M q (n, τ) has been considered for many years in projective geometry, and a survey with the known results is given in [17]. Recently, there has been lot of activity and the question has been almost completely solved [32,33,37].…”

Section: Construction Of Optimal Cooling Codesmentioning

confidence: 99%

Cooling Codes: Thermal-Management Coding for High-Performance Interconnects

Chee

Etzion

Kiah

et al. 2018

IEEE Trans. Inform. Theory

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“…The determination of A 2 (v, 6; 3) for v ≡ 2 (mod 3) was achieved more than 30 years later in [14] and continued to A 2 (v, 2k; k) for v ≡ 2 (mod k) and arbitrary k in [34]. Besides the parameters of A 2 (8 + 3l, 6; 3), for l ≥ 0, see [14] for an example showing A 2 (8, 6; 3) ≥ 34, no partial spreads exceeding the lower bound from Theorem 12 are known.…”

Section: Upper Bounds For Partial Spreadsmentioning

confidence: 99%

Asymptotic Bounds for the Sizes of Constant Dimension Codes and an Improved Lower Bound

Heinlein

Kurz

2017

Coding Theory and Applications

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We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance d = 4, dimension k = 3 of the codewords for all field sizes q, and sufficiently large dimensions v of the ambient space, that exceed the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.

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“…From the work of Segre in 1964 [29,§VI] we know that k-spreads exist if and only if k divides v. Upper bounds for the size of a partial k-spreads are due to Beutelspacher [4] and Drake & Freeman [10] and date back to 1975 and 1979, respectively. Starting from [23] several recent improvements have been obtained. Currently the tightest upper bounds, besides k-spreads, are given by a list of 21 sporadic 1-parametric series and the following two theorems stated in [24]:…”

Section: An Improvement Of the Johnson Bound For Constant Dimension Smentioning

confidence: 99%

On the Lengths of Divisible Codes

Kiermaier

Kurz

2020

IEEE Trans. Inform. Theory

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In this article, the effective lengths of all q r -divisible linear codes over F q with a non-negative integer r are determined. For that purpose, the S q (r)-adic expansion of an integer n is introduced. It is shown that there exists a q r -divisible F q -linear code of effective length n if and only if the leading coefficient of the S q (r)-adic expansion of n is non-negative. Furthermore, the maximum weight of a q r -divisible code of effective length n is at most σ q r , where σ denotes the cross-sum of the S q (r)-adic expansion of n.This result has applications in Galois geometries. A recent theorem of Nȃstase and Sissokho on the maximum sizes of partial spreads follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.1 By [31, Th. 1], for ∆ = p e d with p the characteristic of the base field F q and p d, each full-length ∆divisible F q -linear code is the d-fold repetition of a p e -divisible F q -linear code.

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Improved upper bounds for partial spreads

Cited by 29 publications

References 20 publications

Cooling Codes: Thermal-Management Coding for High-Performance Interconnects

Cooling Codes: Thermal-Management Coding for High-Performance Interconnects

Asymptotic Bounds for the Sizes of Constant Dimension Codes and an Improved Lower Bound

On the Lengths of Divisible Codes

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