Andy Liu

Two Applications of a Hamming Code

Andy Liu (al3@ualberta.ca) has a doctorate in mathematics and a graduate diploma in elementary education. His research interest spans combinatorics, geometry, mathematics recreation, and mathematics education. He is at present on the Board of Governors of the MAA, and serves as co-editor of the Problem Corner in Math Horizons. He was the Deputy Leader of the USA IMO team from 1981 to 1984, and won a Tepper Haimo Award in 2004.

In Search of a Missing Link: A Case Study in Error-Correcting Codes

The topic of error-correcting codes has become a standard item in the undergraduate curriculum in discrete mathematics. Most textbooks, see for example [7], give as a first example of error correcting codes the Alt code [2]. The next example is usually the fa? mous code due to Golay [3] and Hamming [4]. We will give a description of each later. For now, it suffices to say that the Hamming-Golay code is far more sophisticated than the Alt code. What incisive insight had led these two founding fathers of information science to their magnificent result? In this paper, we attempt to find the missing link, namely, a sequence of plausible reasoning that could eventually lead a lesser mortal to the same discovery. Let us introduce the topic of error-correcting codes in the following way. Suppose you have a transmitter which sends fifteen binary digits at a time. Now and then a 0 may be changed into a 1 or vice versa, but at most one of the fifteen digits may be affected. This may cause misunderstanding that can have serious consequences. Suppose transmission errors occur very rarely. Then it is not necessary to correct them on the spot, as long as they can be detected so that retransmission can be requested. This is the complementary topic of error detecting codes. The parity-check code, found in mathematical folklore, detects single transmission errors as follows. In our fifteen-digit transmitter, as many as fourteen digits can be used to convey messages. If the total number of ls used is even, the last digit is chosen to be 0. If the total is odd, then the last digit is chosen to be 1. In either case, the total number of ls in the encoded 15-digit message is supposed to be even. This additional digit is called a parity-check digit. If a digit is reversed during transmission, then the total number of ls in the received message will be odd. This will indicate that retransmission is necessary. We define the efficiency of a code as the ratio of the number of message digits to the total number of digits. Note that the efficiency of this code is ||, which cannot possibly be higher since at least one digit must be used for checking. Suppose single transmission errors occur frequently. Then we may be requesting retransmission back and forth, a highly undesirable situation. Thus we want a code that not only detects the occurrence of single transmission errors, but can actually correct them on the spot. The Alt Code uses only five of the fifteen digits for conveying messages. The last ten digits are just the first five repeated twice. Since each of the five message digits appears three times, its correct value can be determined by the simple majority rule, or two-out-of-three. This simplistic code has the lowly efficiency of ^.

A Better Angle From Outside

This old chestnut has been traced back to at least as early as 1922-23. See [2] and [4] and the references in [lb], [le], and [lh]. In this note, we present three other similar problems. In each case, four angles are given and you are asked to compute /-CAD. Apart from the superficial resem blance in their statements, these four also admit elegant geometric solutions with a common theme. The readers are urged to have a go at them before reading on. Problem 2 was composed for the 1989-90 Alberta High School Mathematics Competition. Only one student got a complete solution, after much tedious trigonometric computation. Actually, there is a very neat trigonometric solution, which points the way to the main theme of this article.

Polyomino Number Theory (II)

Polyominoes are connected plane figures formed of joining unit squares edge to edge. We have a monomino, a domino, two trominoes named I and V, five tetrominoes named I, L, N, O and T, respectively, and twelve pentominoes (a registered trademark of Solomon W. Golomb) named F, I, L, N, P, T, U, V, W, X, Y and Z respectively.

The enumeration problem for color critical linear hypergraphs

Abstract Denote by S ( m , n , r ) the number of non-isomorphic r -critical linear n -graphs on m vertices. It is shown that for n ≥ 3, r ≥ 3 there exists a constant c > 1 depending only on n and r such that S ( m , n , r ) > c m for all sufficiently large m .

Area and Dissection

We shall assume that the readers are familiar with simple terms such as triangles, squares, rectangles and parallelograms, simple concepts such as congruence, similarity, convexity and parallelism, and simple results such as the sum of the angles of a triangle being 180◦. On the other hand, we will make precise the assumptions about area.

Sample GAME Math Unfair Games

The two players take turns, each placing one counter of her color on any vacant square on the board. The objective is for each player to have her three counters in a row, a column or a diagonal.

“From Earth to Moon” Sample Contest I

Nicolas and his son and Peter and his son were fishing. Nicolas and his son caught the same number of fish while Peter caught three times as many fish as his son.

Sample SNAP Math Fair Projects

Each half of the square in Figure 7.1 is a right isosceles triangle constructed with three of the pieces. Find two other solutions.

Three Sample Projects: Counting Problems

Among the Forty Thieves, the Chief Thief is ranked 0, the first mate is ranked 1, and so on down to the bottom mate who is ranked 39.