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Your rating has been recorded. Write a review Rate this item: 1 2 3 4 5. Preview this item Preview this item. Series: Applied and numerical harmonic analysis. A well-established and still highly relevant branch of mathematics, the theory of error-correcting codes is concerned with reliably transmitting data over a 'noisy' channel. Despite its frequent use in a range of contexts--the first close-up pictures of the surface of Mars, taken by the NASA spacecraft Mariner 9, were transmitted back to Earth using a Reed-Muller code--the subject contains interesting problems that have to date resisted solution by some of the most prominent mathematicians of recent decades.
Employing SAGE--a free open-source mathematics software system--to illustrate their ideas, the authors begin by providing background on linear block codes and introducing some of the special families of codes explored in later chapters, such as quadratic residue and algebraic-geometric codes.
Also surveyed is the theory that intersects self-dual codes, lattices, and invariant theory, which leads to an intriguing analogy between the Duursma zeta function and the zeta function attached to an algebraic curve over a finite field. Finally, some of the more mysterious aspects relating modular forms and algebraic-geometric codes are discussed.
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Thus the tetracode words associated in this way to these patterns are the odd men out for the tets. Lemma 77 Conway, [CS1], Chap. These are all possible 6-sets in the shufe labeling for which the odd men out form a part in the sense that and odd man out?
There are 11 hexads with total 21 and none with lower total. There are 11 hexads with total 45 and none with higher total.
These facts will help us play the game mathematical blackjack in Sect. It is a self-dual 4, 2, 3 code. Here is Conways tetracode construction of the C The row score is the sum of the elements in that row.
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The column score is the sum of the elements in that column. Of course, all computations are in GF 3. Lemma 80 Conway An array c is in C 12 if and only if the common score of all four columns equals the negative of the score of the top row; pr c T. There are several facts one can derive from this construction. There are codewords of weight 6, codewords of weight 9, 24 codewords of weight 12, and codewords total.
It is the sup- port of exactly two codewords of weight 9 in C Lemma 81 For each weight 6 codeword c in C 12 , there is a weight 12 codeword c. Proof The support of the codewords of weight 6 form a S 5, 6, 12 Steiner system. Therefore, to any weight 6 codeword c, there is a codeword c. Remark 3 Although we shall not need it, it appears that for each weight 9 codeword c in C 12 , there is a weight 12 codeword c. Dene the label of the rst top row to be 0, of the second row to be 1, and of the bottom row to be 1.
Example 82 Associated to the colcol pattern 1 1 1. There is one, and this section describes it. The views from each of the three points at innity in the shufe labeling is given in the following tables. What is remarkable about it is that a winning strategy, discovered by Conway and Ryba [CS2] and [KR], depends on knowing how to determine hexads in the Steiner system S 5, 6, 12 using the shufe labeling.
Divide the 12 cards into two piles of 6 to be fair, this should be done randomly. Each of the 6 cards of one of these piles are to be placed face up on the table. The remaining cards are in a stack which is shared and visible to both players.
If the sum of the cards face up on the table is less than or equal to 21, then no legal move is possible, so you must shufe the cards and deal a new game. Players alternate moves. A move consists of exchanging a card on the table with a lower card from the other pile. The player whose move makes the sum of the cards on the table under 21 loses. Proposition 84 Ryba For this Steiner system, the winning strategy is to choose a move which is a hexad from this system. This result is proven in [KR].
If you are unfortunate enough to be the rst player starting with a hexad from S 5, 6, 12 , then, according to this strategy and properties of Steiner systems, there is no winning move. In a randomly dealt game there is a probability of In other words, we have the following result. Example 86 Initial deal: 0, 2, 4, 6, 7, The total is The pattern for this deal is 3. No combinations of choices will yield a tetracode odd men out, so this deal is not a hexad. First player replaces 7 by 5: 0, 2, 4, 5, 6, The total is now Note that this is a square in the picture at 1.
Second player replaces 11 by 7: 0, 2, 4, 5, 6, 7. Interestingly, this 6-set corresponds to the pattern. However, it has column distribution 3, 1, 2, 0, so it cannot be a hexad. First player replaces 6 by 3: 0, 2, 3, 4, 5, 7. Note that this is a cross in the picture at 0. Cards total First player wins. The total went from 30 to Is this really a hexad? SAGE sage: M. There is no winning move, so make a random legal move.
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Suppose that player 2 replaced the 11 by a 9. The total went from 27 to You have now won. Shuffle the deck and redeal. Suppose that 11 horses are racing and you must bet on the results of all 11 each bet can be a win, a second place nish, or a third place nish.