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

Clearly require at least 2 XORs

x0 x2 x1 x0+x1+x2

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Let’s consider a simple problem:

How many additions are required when calculating x3 x0 x2 x1

x0+x1+x2 x0+x1+x3 x0+x2+x3 x1+x2+x3 x0+x1

Does this calculation require at least 6 XORs?

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Let’s consider a simple problem:

…… these kind of calculations is the encoding process

• f punctured Hadamard codes and the corresponding

matrices are the generator matrices of punctured Hadamard codes.

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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

• 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

• btain.
• In this paper, we show the exact lower bound of the encoding
• f Hamming codes and Hadamard codes, which are the most

fundamental codes in coding theory.

*Motivations

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Introduction

• Logical circuit can be represented as a directed

graph in which all nodes are with in-degree 0 or 2.

*Logical circuits

Circuit size = number of gates = 2

x0 x2 x1 x0+x1+x2

Input nodes have in- degree 0 Gates have in- degree 2

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Introduction

• Example: The generator matrix of (7,3) Hadamard

codes

*Hadamard codes

consists of all non-zero vectors

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Introduction

• Example: The generator matrix of (8,4) punctured

Hadamard codes

*Punctured Hadamard codes

consists of all columns with odd weight

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Introduction

• The (extended) Hamming codes and (punctured)

Hadamard codes are dual codes. Example: (7,4) Hamming codes and (7,3) Hadamard codes.

*Hamming codes

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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

x0 x2 x1

x0+x1+x2 Input nodes Hidden node Output node Message bits Intermedia result

x0+x1

Parity bit

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.

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• Theorem 1. A (2k-1, k) Hadamard code requires at

least 2k-k-1 XORs in encoding process based on the generator matrix.

*Hadamard codes

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• Lemma 1. There is at least 1 hidden node in any

encoding circuit of punctured Hadamard codes.

*Punctured Hadamard codes

all odd weight

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x0+x1 even weight

• Lemma 2. If there is an encoding circuit of (2k, k+1)

punctured Hadamard codes with m hidden nodes, then there is an encoding circuit of (2k-1, k) punctured Hadamard codes with at most m-1 hidden nodes.

*Punctured Hadamard codes

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The idea of the proof of Lemma 2.

*Punctured Hadamard codes

x3 x0 x2 x1

x0+x1+x2 x0+x1+x3 x0+x2+x3 x1+x2+x3 x2+x3

x2 x0 x2 x1

x0+x1+x2 x0+x1+x2 x0+x2+x2 x1+x2+x2 x2+x2

Let x3=x2

(8,4) punctured Hadamard codes (4,3) punctured Hadamard codes

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• Lemma 3. The number of hidden nodes in any

encoding circuit of (2k, k+1) punctured Hadamard codes is at least k-1.

• Theorem 2. A (2k, k+1) punctured Hadamard code

requires at least 2k-2 XORs in encoding process based on generator matrix.

✴ We also devise encoding algorithms achieving the

lower bounds in Theorems 1 and 2.

*Punctured Hadamard codes

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• 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 viM with a vector vi of length i. Then there exists an algorithm to compute vjMT in a(M) + j − i arithmetic operations.

*(Extended) Hamming codes

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• Hadamard codes
• Generator matrix

G=[I A]

• Encoding: xA
• The least XORs

required in xA (Theorem 1)

*(Extended) Hamming codes

• Hamming codes
• Generator matrix

G’=[I’ AT]

• Encoding: x’AT
• The least XORs

required in x’AT (Theorem 3)

*The property of dual code

By contradiction

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• Theorem 4. A (2k-1, 2k-k-1) Hamming code requires

at least 2k+1−3k−2 XORs in encoding process based on the generator matrix.

• Theorem 5. A (2k, 2k-k-1) extended Hamming

codes requires at least 2k+1−2k−4 XORs in encoding process based on the generator matrix.

*(Extended) Hamming codes

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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 [m0 … m3], let x=[0 0 0 m0 0 m1 m2 m3]. p=[(0)3 (1)3 … (7)3]xT=x0(0)3+…+x7(7)3.

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The encoding algorithms of Hamming codes

[p1 p2 p3]T=(x0+x4)(0)3+(x1+x5)(1)3+… +(x3+x7)(3)3+(x4+x5+x6+x7)[1 0 0]T, Further, we establish

(a) [p2 p3]T=(x0+x4)(0)2+…+(x3+x7)(3)2 (b) p1=x4+x5+x6+x7

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The encoding algorithms of Hamming codes

• Thus, [p2 p3]T can be obtained recursively by

applying the same approach on p.

The graph of this recursive approach

This circuit requires 5 XORs which achieves the lower bound in Theorem 4

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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

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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.

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—Zhengrui Li

Thanks!