Thermal-Management Coding for High-Performance Interconnects

Transcription

1 Cooling Codes Thermal-Management Coding for High-Performance Interconnects Han Mao Kiah, Nanyang Technological University, Singapore Joint work with: Yeow Meng Chee, Nanyang Technological University Tuvi Etzion, Technion, Israel Institute of Technology Alexander Vardy, University of California, San Diego

2 DSM Bus Communication Sender Receiver s are connected via wires. Each wire has two states to represent a bit of information. Problem When a wire switches state, or when there is a state transition, the wire heats up.

3 Minimizing Switching Activity Previous work focus on reducing the number of state transitions. Encoding techniques: Bus- Invert (Stan and Burleson 995) Thermal Spreading (Wang et al. 27) Information theoretic analysis (Sotiriadis et al. 23, Koch et al. 29) Sender Receiver Bus- Invert Introduces one redundant wire. Chooses to send x or its complement x so that the number of state transitions is at most n/2 for n wires.

4 Controlling Peak Temperature How? Avoiding state transitions on the hottest wires. Sender Receiver Why? To handle anomalous events. Source signals are not usually uniformly distributed.

5 Key Features of Coding Scheme Consider a bus comprising n wires. Property A(t): Every transmission does not cause state transitions on the t hottest wires. Property B(w): Every transmission causes state transitions on at most w wires. Sender Receiver Property C(e): Correct up to at most e transmission errors. n = 6 t = 2 w = 2

6 Property A(t): Every transmission does not cause state transitions on the t hottest wires. This talk Property B(w): Every transmission causes state transitions on at most w wires. Property C(e): Correct up to at most e transmission errors. Code constructions that achieve Property B(w) only - Optimal Property A(t) only Optimal redundancy Property A(t) + Property B(w) only Asymptotically optimal redundancy Adaptive coding schemes. In the full paper on arxiv, More constructions that achieve Property A(t) + Property C(e) only Property B(w) + Property C(e) only Property A(t) + Property B(w) + Property C e Nonadaptive coding schemes

7 Adaptive + Differential Coding Sender Receiver Suppose Sender wants to communicate. Codeword Codeword Buffer We change the states to. Receiver adds the current state vector to the vector in the buffer to retrieve the message. Receiver updates the buffer.

8 Differential Coding and Property B Sender Receiver Property B(w): Every transmission causes state transitions on at most w wires. A code satisfies Property B(w) if and only if Codeword Codeword wt x w for codewords x. In other words, the optimal code that satisfies Property B w is Buffer n, w = x wt x w.

9 Differential Coding and Property A Sender Receiver Property A(t): Every transmission does not cause state transitions on the t hottest wires. A code satisfies Property A(t) if and only if Codeword??????????? Codeword??????????? for all codewords x, for all subsets S of size t, x J = for all i S. Buffer This implies that x =!! t =

10 Cooling Codes Property A(t): Every transmission does not cause state transitions on the t hottest wires. Instead of vectors, we consider codesets An (6,2)-cooling code C O =,,,,,,, C P =,,,,,,, C Q =,,,,,,, C R =,,,,,,, C S =,,,,,,, C T =,,,,,,, C U =,,,,,,, C V =,,,,,,, C W =,,,,,,. An (n, t)- cooling code is a collection of M codesets C O, C P,, C Z such that C O, C P,, C Z are disjoint subsets. for all codesets C J, for all subsets S of size t, there exist a word x C J x ] = for all j S.

11 Cooling Codes Property A(t): Every transmission does not cause state transitions on the t hottest wires. Consider the codeset C O and S =,2. An (6,2)-cooling code C O =,,,,,,. An (n, t)- cooling code is a collection of M codesets C O, C P,, C Z such that C O, C P,, C Z are disjoint subsets. for all codesets C J, for all subsets S of size t, there exist a word x C J x ] = for all j S, or, x =. _`

12 Cooling Codes and Related Codes Definition An (n, t)- cooling code is a collection of M codesets C O, C P,, C Z such that C O, C P,, C Z are disjoint subsets. Coding for Stuck- At Cells for all codesets C J, for all subsets S of size t, there exist a word x C J x ] = for all j S, or x ` =. Additional requirement: x ` = z for all patterns z of length t Cooling codes require significantly lesser redundancy Dumer (989) constructed a special class of codes for stuck- at- cells Write- Once Memories (WOM) codes Different requirement: x ` = Different requirement: S need not be a subset of size t Cooling codes can be used to construct WOM codes. Explicit Constructions of Finite- Length WOM Codes ISIT, 3 Jun Fri, 2:3pm, Room: K5

13 Upper Bound on the Code Size Sender Receiver Lemma A (n, t)- cooling code has at most 2 ghi codesets. proof Corollary Since If < there t < are n no, state an (n, transition t)- cooling on t wires, code has no information most 2 ghi is transmitted. codesets. Therefore, at most n t bits of information can be transmitted. Property A(t): Every transmission does not cause state transitions on the t hottest wires.

14 Lower Bound on the Code Size Sender Receiver Lemma A (n, t)- cooling code has at most 2 ghi codesets. Theorem If t + n/2, there is an (n, t)- cooling code of size at least 2 ghiho codesets. In other words, construction is optimal in terms of redundancy. Property A(t): Every transmission does not cause state transitions on the t hottest wires.

15 Spreads and Partial Spreads A partial τ- spread of F P g is a collection of τ- dimensional subspaces V O, V P,, V Z of F P g such that V J V ] = {} for all i j. If V J = F P g, then V O, V P,, V Z is called a τ- spread. A 3-spread of F P T V O = span,,, V P = span,,, V Q = span,,, V R = span,,, V S = span,,, V T = span,,, V U = span,,, V V = span,,, V W = span,,.

16 Spreads yields Cooling Codes Definition Theorem Let V O, V P,, V Z be a partial (t + )- spread. Set V J = V J {}. Then V O, V P,, V Z forms an (n, t)- cooling code. V O = span,, {}, V P = span,, {}, V Q = span,, {}, V R = span,, {}, V S = span,, {}, V T = span,, {}, V U = span,, {}, V V = span,, {}, V W = span,, {}. A partial τ- spread of F P g is a collection of τ- dim subspaces V O, V P,, V Z of F P g such that V J V ] = {} for all i j. Definition An (n, t)- cooling code is a collection of M codesets C O, C P,, C Z such that C O, C P,, C Z are disjoint subsets. for all codesets C J, for all subsets S of size t, there exist a word x C J x ] = for all j S. A 3-spread of F P T yields a (6,2)-cooling code

17 proof Spreads yields Cooling Codes Theorem Let V O, V P,, V Z be a (t + )- partial spread. Set V J = V J {}. Then V O, V P,, V Z forms an (n, t)- cooling code. Definition A partial τ- spread of F P g is a collection of τ- dim subspaces V O, V P,, V Z of F P g such that V J V ] = {} for all i j. Definition An (n, t)- cooling code is a collection of M codesets C O, C P,, C Z such that C O, C P,, C Z are disjoint subsets. for all codesets C J, for all subsets S of size t, there exist a word x C J x ] = for all j S. V J V ] = for i j follows from definition.

18 proof Spreads yields Cooling Codes Theorem Let V O, V P,, V Z be a (t + )- partial spread. Set V J = V J {}. Then V O, V P,, V Z forms an (n, t)- cooling code. Definition A partial τ- spread of F P g is a collection of τ- dim subspaces V O, V P,, V Z of F P g such that V J V ] = {} for all i j. Definition An (n, t)- cooling code is a collection of M codesets C O, C P,, C Z such that C O, C P,, C Z are disjoint subsets. for all codesets C J, for all subsets S of size t, there exist a word x C J x ] = for all j S. Consider the codeset V O and S =,2. V O = span,,. Want to find x V O such that x ] = for all j S, or, x O = x P =.

19 proof Spreads yields Cooling Codes Consider the codeset V O and S =,2. V O = span,,. Want to find x V O such that x O = x P =. (*) Let K be the collection of vectors that satisfy (*). Suppose x, y K. i.e. x V O such that x O = x P =. y V O such that y O = y P =. Then x + y K {}. In other words, K {} is a vector subspace of V O.

20 proof Spreads yields Cooling Codes Consider the codeset V O and S =,2. V O = span,,. Want to find x V O such that x O = x P =. (*) Let K be the collection of vectors that satisfy (*). In fact, K {} is the kernel of the map φ that projects V O onto the coordinates at S. φ: V O F P` Since V O has dimension three and its image has dimension two, its kernel K {} must be nontrivial. So, there exists nonzero x that satisfies (*).

21 Cooling Codes - Property A Theorem Let V O, V P,, V Z be a (t + )- partial spread. Set V J = V J {}. Then V O, V P,, V Z forms an (n, t)- cooling code. Theorem (Classic + Etzion and Vardy 2) Let τ n/2. There is a τ- partial spread of size at least 2 gh. Definition A partial τ- spread of F P g is a collection of τ- dim subspaces V O, V P,, V Z of F P g such that V J V ] = {} for all i j. Definition An (n, t)- cooling code is a collection of M codesets C O, C P,, C Z such that C O, C P,, C Z are disjoint subsets. for all codesets C J, for all subsets S of size t, there exist a word x C J x ] = for all j S. Theorem If t + n/2, there is an (n, t)- cooling code of size at least 2 ghiho codesets. The construction is optimal in terms of redundancy. (Dumer 989) There exist efficient encoding and decoding methods for spreads.

22 Property A and B Property A(t): Every transmission does not cause state transitions on the t hottest wires. Property B(w): Every transmission causes state transitions on at most w wires. Let n = 286, t = 3, w = 27. Consider a 4- partial spread of F P PVT of size 2 PRT. Let V O, V P,, V P ƒ be the vector spaces. Let V J = V J {} be the codesets. Choose a codeset V J. For any subset S of size 3, the kernel K of the projection φ: V J F P` has dimension of at least. We puncture K at the coordinates in S to obtain a By Plotkin bound, subspace K of dim and length 256. there exists nonzero x K with wt x 27.

23 Property A and B Property A(t): Every transmission does not cause state transitions on the t hottest wires. Property B(w): Every transmission causes state transitions on at most w wires. Let n = 286, t = 3, w = 24. Consider a 4- partial spread of F P PVT of size 2 PRT. Let V O, V P,, V P ƒ be the vector spaces. Let V J = V J {} be the codesets. Choose a codeset V J. For any subset S of size 3, the kernel K of the projection φ: V J F P` has dimension of at least. We puncture K at the coordinates in S to obtain a Checking codetables.de, subspace K of dim and length 256. there exists nonzero x K with wt x 24.

24 Property A and B Theorem Suppose that t + r (n + s)/2. If i. there is an [n, s, w + ]- linear code, and ii. an n t, r, w + - linear code does not exist, then there is a code that satisfies Property A(t) and B(w) of size 2 ghih. Property A(t): Every transmission does not cause state transitions on the t hottest wires. Property B(w): Every transmission causes state transitions on at most w wires.

25 Cooling Codes: Thermal-Management Coding for High-Performance Interconnects Full Paper Available at Other Results: Construction of (n, t)- cooling code for t + > n/2. Construction of codes that satisfy Property A(t) and B(w) using Baranyai s theorem, concatenation Construction of codes that satisfy additional Property C(e)