Codes

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Codes the Hat Game

• Troy Lynn Bullock
• John H. Reagan High School, Houston ISD
• Shalini Kapoor
• McArthur High School, Aldine ISD
• Faculty Mentor Dr. Tie Liu
• Graduate Assistant Neeharika Marukala

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Outline

• An introduction to communication systems
• Error correction codes
• The hat game
• Lesson plan

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Introduction

• Communications touches the lives of everyone in
  many ways.
• Lets look at some applications of communications
  in this information age!

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An Information Age

• The Internet

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An Information Age

• Deep-space communication

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An Information Age

• Satellite broadcasting

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An Information Age

• Cell phone and modem

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An Information Age

• Data storage

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Basic Communication System
Information Source
Destination
Bits
Bits
Transmitter
Receiver
Distortion
Waveform
Waveform
Communication Channel
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Communication Channel

• Introduce distortion to the transmit signal
• As a result, some bits are flipped at the
  receiver (e.g., 0?1 or 1?0)
• Which bits will be flipped are random/unpredictabl
  e
• Random bit flipping conveys false information to
  the destination and is bad for communication
• Solution Error Correction Codes

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Repetition Codes

• Consider using bit 0 to represent 0 and 1 to
  represent 1
• If the bit is flipped, then we have no idea which
  bit was sent
• Now consider using three bits 000 to represent
  0 and 111 to represent 1
• If only one of the three bits is flipped, we can
  still make out which bit was sent by looking at
  the majority of the bits
• More errors can be corrected by making more
  repetitions
• Research question Can we find codes that are
  more efficient than repetition codes?

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Coding Efficiency
Can we do better?
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Coding Theory

• A branch of modern mathematics
• With deep connections to
• Theory of finite field
• Algebraic geometry
• Number theory
• Combinatorics
• Algorithm
• Complexity theory
• Information Theory
• With applications from deep-space communications
  to consumer electronics
• A perfect example on how good mathematics can
  significantly impact our daily life

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Achieving Immortalities
Richard Hamming Irvine Reed Gustave
Solomon Elwyn Berlekamp Claude Berrou Robert
Gallager
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The Hat Game

• The Setup One team of three contestants are in a
  room.
• A red or blue hat is randomly put on each
  contestant
• each contestant can see the hats of
  everyone else
• but not his/her own.
• The Game Each contestant must (simultaneously)
• 1. Guess the color of his/her hat,
• 2. Or pass.
• To Win At least one contestant guessed
  correctly,
• and no one guessed incorrectly.
• The team can confer on a strategy beforehand.

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What Strategy Can Be Used?

• A Naïve Strategy
• Pick a team captain.
• The captain guesses red/blue randomly.
• everyone else passes.
• This strategy wins 50 of the time.

CAN WE DO BETTER??
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A Better Strategy

• Each contestant does
• If the other two hats are different colors, pass.
  
• If the other two hats are the same color, guess
  the opposite color.

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Analysis

• This strategy wins of the time!
  

HATS GUESSES WIN?
000 111 no
100 1xx yes
010 x1x yes
001 xx1 yes
110 xx0 yes
101 x0x yes
011 0xx yes
111 000 no
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Lessons Learned

• Its OK to make a mistake. But when we make
  mistakes, its better to make mistakes together
  as a team.
• When lacking evidence, its good to keep quiet
  for the team.

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Recording Sheet for Hat Game
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Geometric Interpretations

• For n3 players

110
001
000
111
011
010
100
101
Bad sequence
Good sequence
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Geometric Interpretations

• For n2k-1 players, use Hamming Codes as bad
  sequences

Hamming Ball
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Perfect Codes

• For n2k-1, all possible 2n binary sequences can
  be partitioned into Hamming balls of radius 1
• Since the Hamming balls are non-overlapping,
  Hamming codes can correct any single bit flip at
  the minimum redundancy
• For k2, Hamming codes are the same as repetition
  codes
• For kgt2, Hamming codes are much more efficient
  than repetition codes

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Coding Efficiency
Hamming Codes
Repetition Codes
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What about n?2k-1?

• Still need to cover all binary sequences using
  Hamming balls of radius 1
• The Hamming balls may have to overlap

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What about n?2k-1?

• What are the optimum choices for the bad
  sequences?
• Answers are known only for n38 despite the
  effort of many famous mathematicians
• A perfect challenge for kids to try a world-class
  open problem with strong engineering implications!

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What Can the Kids Learn?

• Permutations/Combinations
• Probability
• Percents
• Cooperative Learning
• Team Work
• Decision Making

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Lesson Plan
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Lesson Plan cont
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Sample Questions
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Acknowledgements

• I would like to acknowledge E3 for giving me the
  opportunity to experience new and different
  things. Also, Dr. Liu and Neeharika Marukala for
  enhancing my knowledge in Engineering so that I
  will be able to bring future students to Texas
  AM University that will major in ENGINEERING
  perhaps Electrical Engineering.
• Also, I would like to thank National Science
  Foundation (NSF), Nuclear Power Institute (NPI),
  Texas Workforce Commission (TWC), and Chevron.
  The support from these groups have made E3
  Program what it is today.