Self-Complementing Codes (Excess 3, 84-2-1, 2*421) - bcd

Question

I'm having a topic here which is from "Number Systems" in the subject of "Introduction to Computer Organisation & Architecture"
Then i came across this topic,"Self complementing Codes"
There are 3 parts of it which are as follows:
i)Excess-3 (I understand this part as it requires us to add 3 to BCD)
ii)84-2-1 (I don't understand)
iii)2*421 (I don't understand)
I hope someone could explain how the part ii & iii works.
Thanks alot.

Answer 1

I think this part of "Digital Design" book from Morris Mano will answer your question:
BCD and the 2421 code are examples of weighted codes. In a weighted code, each bit
position is assigned a weighting factor in such a way that each digit can be evaluated by
adding the weights of all the 1’s in the coded combination.
Four Different Binary Codes for the Decimal Digits
_____________________________________________________________
Decimal BCD 2421 Excess‐3 8, 4, -2, -1
Digit 8421
_____________________________________________________________
0 0000 0000 0011 0000
1 0001 0001 0100 0111
2 0010 0010 0101 0110
3 0011 0011 0110 0101
4 0100 0100 0111 0100
5 0101 1011 1000 1011
6 0110 1100 1001 1010
7 0111 1101 1010 1001
8 1000 1110 1011 1000
9 1001 1111 1100 1111
_____________________________________________________________
1010 0101 0000 0001
Unused 1011 0110 0001 0010
bit 1100 0111 0010 0011
combi- 1101 1000 1101 1100
nations 1110 1001 1110 1101
1111 1010 1111 1110
The 2421, the excess‐3 and the 84-2-1 codes are examples of self‐complementing codes. Such
codes have the property that the 9’s complement of a decimal number is obtained
directly by changing 1’s to 0’s and 0’s to 1’s (i.e., by complementing each bit in the pattern). For example, decimal 395 is represented in the excess‐3 code as 0110 1100 1000.
The 9’s complement of 604 is represented as 1001 0011 0111, which is obtained simply
by complementing each bit of the code (as with the 1’s complement of binary numbers).
Digital Design-Fifth edition-By Morris Mano

Answer 2

First of all 84-2-1 & 2421 code are "weighted code" and as well as "self-complementing code" both (because the necessary condition for a code to be self-complementing is that the sum of all of its weight must be equal to 9) i.e. 84-2-1(8+4-2-1=9) and, 2421(2+4+2+1=9).
So, constructing binary equivalent of decimal of number you to need to make sure one thing:
The number's code and its 9's complement's code should have complementary relationship (i.e. the number & its 9's complement code are complements of each other)
For example let's take a 84-2-1 System.
Decimal 0=0000(in 84-2-1 system) & (9's complement of 0=9) than 9(in 84-2-1) should have 1111. Hence 0 & its 9's complement i.e. 9 maintain their self-complement relationship.
Let's take another example:
Decimal 1=0111(in 84-2-1 system) & (9's complement of 1=8 ) that implies (84-2-1) equivalent of 8 should be 1000,hence again the number & its complement preserve their self-complementary relationship.
Similarly, as in 2421 system, all you need to construct the code by making sure that the number and its 9's complement should have maintained their self-complementing relationship.