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This paper shows the existence of a ternary triorthogonal code and a comma-free ternary triorthogonal code. Some theorems are proved for their existence and some algorithms for generating these codes are developed. For a dictionary of size $D = 3^{n + 1} $ and word length $L = 3^n $, the former code exists for any positive integer n and the latter code exists for $n > 1$. The ternary elements $\{ 0,1,2\} $ used in the codes are isomorphic, under the one-to-one correspondence, to the triphase vectors $\{ e^{(2\pi i/3 )r} ,r = 0,1,2\}$. The modulo 3 addition of $\{ 0,1,2\} $ and the triphase vector multiplication are also isomorphic under the one-to-one correspondence.

MANY students find it difficult to get very excited about formal proofs of theorems that they already believe intuitively. Properly developing the field axioms and the usual theorems that follow requires several class meetings. To the more mathematically mature, the process of justifying many operations that have previously been taken for granted is one of the most satisfying of all activities in mathematics. The care with which the field axioms were chosen is elegantly demonstrated as proofs are developed for theorems that have previously been accepted because they appealed to the intuition. Many of our students, however, fail to appreciate the beauty of these proofs. Perhaps one of the reasons that they remain unmoved is that they see little application of the field properties other than to facts that seem intuitively obvious to them.

A model is presented which, when modulo addition is used, can be used to derive multiplication correction schemata for operands expressed in either the radix complement or the diminished radix complement code.

In recent years several papers dealing with the mathematical theory as well as the technical applications of Walsh functions have been published. One method of defining Walsh functions is by using the Rademacher functions and the multiplication law of Walsh functions which is in fact a binary addition modulo 2 (no carry). This letter, however, describes in detail a method of defining Walsh functions by using orthogonal code blocks or the so-called Hadamard matrices. Some advantages of this method are its simplicity and its straightforward hardware implementation.

Modulo arithmetic has been described as a “clock” arithmetic. A modulo system, being a finite system, provides an excellent setting for the study of arithmetical concepts. Most students readily grasp the concepts of addition and subtraction through the use of tables. Some of the slower students have difficulty with the multiplication and division concepts. The tables can be constructed through the use of “clocks” on which the elements have been represented. It is also possible to illustrate the concepts of modulo, or clock, arithmetic on the more familiar number line.