On the Exact Lower Bounds of Encoding Circuit Sizes of Hamming codes - PowerPoint PPT Presentation

On the Exact Lower Bounds of Encoding Circuit Sizes of Hamming codes and Hadamard codes Zhengrui Li, Sian-Jheng Lin and Yunghsiang S. Han presented by Zhengrui Li ISIT2020

Let’s consider a simple problem: How many additions (XORs) are required when calculating x 0 +x 1 +x 2 Clearly require at least 2 XORs x 0 x 1 x 2 2

Let’s consider a simple problem: How many additions are required when calculating x 0 +x 1 +x 2 x 0 +x 1 +x 3 x 0 +x 2 +x 3 x 1 +x 2 +x 3 Does this calculation x 0 +x 1 require at least 6 XORs? x 0 x 1 x 2 x 3 3

Let’s consider a simple problem: …… these kind of calculations is the encoding process of punctured Hadamard codes and the corresponding matrices are the generator matrices of punctured Hadamard codes. 4

Outline • Introduction ✴ Motivations ✴ Logical circuits ✴ Hadamard codes & Hamming codes • The Lower Bounds of Encoding Circuits Sizes of Hadamard Codes and Hamming Codes • The Encoding Algorithms which Achieve these Lower Bounds • Conclusions & Future Works

Introduction *Motivations • It is difficult to explore the lower bound of the encoding complexities of most linear block codes. • There are a few asymptotically good linear codes with linear encoders, such as expander codes. For the linear codes with linear encoders, the constant factors hidden in big-O complexity are unknown. Furthermore, the exact lower bound (the number of arithmetic operations) is much harder to obtain. • In this paper, we show the exact lower bound of the encoding of Hamming codes and Hadamard codes, which are the most fundamental codes in coding theory. 6

Introduction *Logical circuits • Logical circuit can be represented as a directed graph in which all nodes are with in-degree 0 or 2. x 0 +x 1 +x 2 Gates have in- Circuit size = degree 2 number of gates = 2 Input nodes have in- degree 0 x 0 x 1 x 2 7

Introduction *Hadamard codes • Example: The generator matrix of (7,3) Hadamard codes consists of all non-zero vectors 8

Introduction *Punctured Hadamard codes • Example: The generator matrix of (8,4) punctured Hadamard codes consists of all columns with odd weight 9

Introduction *Hamming codes • The (extended) Hamming codes and (punctured) Hadamard codes are dual codes. Example: (7,4) Hamming codes and (7,3) Hadamard codes. 10

Outline • Introduction ✴ Motivations ✴ Logical circuits ✴ Hadamard codes & Hamming codes • The Lower Bounds of Encoding Circuits Sizes of Hadamard Codes and Hamming Codes • The Encoding Algorithms which Achieve these Lower Bounds • Conclusions & Future Works

The Lower Bounds of Encoding Circuits Sizes • Example of encoding circuits: (4,3) punctured Hadamard codes x 0 +x 1 +x 2 Output node Parity bit x 0 +x 1 Hidden node Intermedia result Input nodes Message bits x 0 x 1 x 2 Observation: All output nodes are gates which implies the encoding circuit size is the number of parity bits plus the number of the hidden nodes. 12

*Hadamard codes • Theorem 1. A (2 k -1, k) Hadamard code requires at least 2 k -k-1 XORs in encoding process based on the generator matrix. 13

*Punctured Hadamard codes • Lemma 1. There is at least 1 hidden node in any encoding circuit of punctured Hadamard codes. x 0 +x 1 even weight all odd weight 14

*Punctured Hadamard codes • Lemma 2. If there is an encoding circuit of (2 k , k+1) punctured Hadamard codes with m hidden nodes, then there is an encoding circuit of (2 k-1 , k) punctured Hadamard codes with at most m-1 hidden nodes. 15

*Punctured Hadamard codes � The idea of the proof of Lemma 2. x 0 +x 1 +x 2 x 0 +x 1 +x 3 x 0 +x 2 +x 3 x 1 +x 2 +x 3 x 0 +x 1 +x 2 x 0 +x 1 +x 2 x 0 +x 2 +x 2 x 1 +x 2 +x 2 x 2 +x 3 x 2 +x 2 x 0 x 1 x 2 x 3 x 0 x 1 x 2 x 2 (4,3) punctured Hadamard codes (8,4) punctured Hadamard codes Let x 3 = x 2 16

*Punctured Hadamard codes • Lemma 3. The number of hidden nodes in any encoding circuit of (2 k , k+1) punctured Hadamard codes is at least k-1 . • Theorem 2. A (2 k , k+1) punctured Hadamard code requires at least 2 k -2 XORs in encoding process based on generator matrix. ✴ We also devise encoding algorithms achieving the lower bounds in Theorems 1 and 2. 17

*(Extended) Hamming codes • Theorem 3. (Transposition principle): Given an i-by- j matrix M without zero rows or columns, let a(M) denote the minimum number of operations to compute the product v i M with a vector v i of length i . Then there exists an algorithm to compute v j M T in a(M) + j − i arithmetic operations. 18

*(Extended) Hamming codes *The property of dual code • Hadamard codes • Hamming codes • Generator matrix • Generator matrix G=[I A] G’=[I’ A T ] • Encoding: xA • Encoding: x’A T • The least XORs • The least XORs required in xA required in x’A T By contradiction (Theorem 1) (Theorem 3) 19

*(Extended) Hamming codes • Theorem 4. A (2 k -1, 2 k -k-1) Hamming code requires at least 2 k+1 − 3k − 2 XORs in encoding process based on the generator matrix. • Theorem 5. A (2 k , 2 k -k-1) extended Hamming codes requires at least 2 k+1 − 2k − 4 XORs in encoding process based on the generator matrix. 20

Outline • Introduction ✴ Motivations ✴ Logical circuits ✴ Hadamard codes & Hamming codes • The Lower Bounds of Encoding Circuits Sizes of Hadamard Codes and Hamming Codes • The Encoding Algorithms which Achieve these Lower Bounds • Conclusions & Future Works

The encoding algorithms of Hamming codes • Example: (7,4) Hamming codes Given message vectors [ m 0 … m 3 ], let x=[ 0 0 0 m 0 0 m 1 m 2 m 3 ]. p=[(0) 3 (1) 3 … (7) 3 ]x T = x 0 (0) 3 +…+ x 7 (7) 3 . 22

The encoding algorithms of Hamming codes [ p 1 p 2 p 3 ] T =( x 0 + x 4 )(0) 3 +( x 1 + x 5 )(1) 3 +… +( x 3 + x 7 )(3) 3 +( x 4 + x 5 + x 6 + x 7 )[1 0 0] T , Further, we establish (a) [ p 2 p 3 ] T =( x 0 + x 4 )(0) 2 +…+( x 3 + x 7 )(3) 2 (b) p 1 = x 4 + x 5 + x 6 + x 7 23

The encoding algorithms of Hamming codes • Thus, [ p 2 p 3 ] T can be obtained recursively by applying the same approach on p. This circuit requires 5 XORs which achieves the lower bound in Theorem 4 The graph of this recursive approach 24

The encoding algorithms of Hamming codes • Here is the encoding circuit of extended Hamming codes This circuit requires 6 XORs which achieves the lower bound in Theorem 5 25

Conclusions & Future Works • The lower bounds of the encoding circuit sizes of (punctured) Hadamard codes and (extended) Hamming codes are presented. • The encoding algorithms which achieving these lower bounds are proposed to show these lower bounds are tight. ✴ A possible future work is to find the exact lower bounds on the encoding circuit size of other more complex linear error correcting codes. 26

Thanks! —Zhengrui Li