# ternary

In the case of transmission methods, an attempt is made to increase the data transfer rate at the same data rate using multi- level coding( MLC) with coded modulation. For this purpose, one bit or a bit combination is assigned to each of the individual levels.

In the ternary method, the signal elements or code elements are based on three discrete values: a positive Hi level, the 0 level and a negative Lo level.

The ternary system is a three-level system and operates with base 3, i.e. `3^1, 3^2, 3^3, 3^4` etc. The two intermediate states for the decimal numbers 1 and 2 are also given as 1 and 2 in the normal ternary system. This means that the decimal number 0 is also a 0 in the ternary system, the number 1 is also a 1 and the 2 is a 2. The decimal digit 3 is the first power of 3 and therefore a 10 in the ternary system. Correspondingly, the digits 9 as the second power are ternary 100, 27 as the third power 1000, 54 as the fourth 1000. The intermediate digits can only be 0, 1 or 2.

## Decimal numbers and ternary values

Mathematically, the conversion of a decimal number into a ternary number is done by division by 3, where only the remainder is represented. For example, if you convert the digit 17 into the ternary system, you divide the 17 by 3. This results in 5 remainder 2. The last digit of the ternary number is 2. The result 5 is again divided by 3, which results in 1 remainder 2. The second to last ternary digit is also a 2. The result 3 is again divided by 3, which results in 1. The complete ternary number for the decimal 17 is thus: 122.

In the ternary system, a distinction is made between the normal and the balanced ternary system. While the normal ternary system represents the three level values with the digits 0, 1 and 2, the balanced ternary system works with signs. Since the notation can lead to errors, the "-1" is also represented as a one with underscore instead of the sign, or the balanced version is referred to in a footnote. The negative value in the ternary number is subtracted. For a balanced ternary number of 10-1, the result is a decimal number of `1*3^2 + 0*3^1 - 1*3^0 = 9 + 0 -1 = 8`. The advantage of the balanced ternary system is that the ternary numbers can be negated by sign reversal.

Ternary methods include duobinarycoding, AMI coding, HDB coding with HDB3, B6ZS, 3B2T, 4B3T, Paired Selected Ternary (PST) and MLT coding with MLT-3.