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

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

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

An Information Age The Internet

An Information Age Deep-space communication

An Information Age Satellite broadcasting

An Information Age Cell phone and modem

An Information Age Data storage

Basic Communication System Information Source TransmitterReceiver Destination Communication Channel Bits Waveform Distortion

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/unpredictable Random bit flipping conveys false information to the destination and is bad for communication Solution: Error Correction Codes

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?

Coding Efficiency Can we do better?

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

Achieving Immortalities Richard Hamming Irvine Reed & Gustave Solomon Elwyn Berlekamp Claude Berrou Robert Gallager

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.

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??

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.

Analysis This strategy wins of the time! HATSGUESSESWIN? no 1001xxyes 010x1xyes 001xx1yes 110xx0yes 101x0xyes 0110xxyes no

Lessons Learned It’s OK to make a mistake. But when we make mistakes, it’s better to make mistakes together as a team. When lacking evidence, it’s good to “keep quiet” for the team.

Recording Sheet for Hat Game

Geometric Interpretations For n=3 players “Bad” sequence “Good” sequence

Geometric Interpretations For n=2 k -1 players, use Hamming Codes as “bad” sequences Hamming Ball

“Perfect” Codes For n=2 k -1, all possible 2 n 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 k=2, Hamming codes are the same as repetition codes For k>2, Hamming codes are much more efficient than repetition codes

Coding Efficiency Hamming Codes Repetition Codes

What about n≠2 k -1? Still need to “cover” all binary sequences using Hamming balls of radius 1 The Hamming balls may have to overlap

What about n≠2 k -1? What are the optimum choices for the “bad” sequences? Answers are known only for n=3~8 despite the effort of many famous mathematicians A perfect challenge for kids to try a world- class open problem with strong engineering implications!

What Can the Kids Learn? Permutations/Combinations Probability Percents Cooperative Learning Team Work Decision Making

Lesson Plan

Lesson Plan cont…

Sample Questions

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 A&M 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.