These codes are used to "detect" and"correct" errors during a digital transmission.
In this simple example of transmission, the transmitter can only send outcodes from the following set of 7-bit words:
0000000 (0)
0000111 (7)
0011001 (25)
0011110 (30)
0101011 (43)
0101100 (44)
0110010 (50)
0110101 (53)
1001010 (74)
1001101 (77)
1010011 (83)
1010100 (84)
1100001 (97)
1100110 (102)
1111000 (120)
1111111 (127)
When the transmitted words land at the receiver, some of thosecodes may be corrupted.
It is assumed that only 1-bit corruption occurs.
Example: the code 0000000 is transmitted, but it lands at thereceiver corrupted by 1 bit, say 1000000.
The receiver system immediately "detects" the errorbecause this code is not expected, it does not belong to the setof possible 16 codes that had been transmitted.
In addition, the receiver system finds out that the only codediffering by 1 bit and belonging to the above table is 0000000,all other 15 codes of the table differ by more than 1, therefore0000000 GOT to have been the one transmitted, and the receivedcorrupted code 1000000 will be therefore corrected into theoriginal value 0000000.
You may try any 1-bit corrected code and see whether you stillcan detect the original unique code.
Let us say for example the received code equals 1101110: therecan be only one code belonging to the table and differing by 1 bit,precisely 1100110 (102).
All other 15 codes differ by 2 or more bits from the receivedcode.
Whichever code will be corrupted, it will be detected (becausebeing unexpectedly different by any of the 16 possibletransmitted codes) and will be corrected into its original value,which is ONLY one out of the table's 16 codes differing by 1 bit.

ANALOGY WITH DRAWING FROM THE 16 CODES LIKE LOTTERY TICKETS

In this case a system of 16 codes as above is purchased.
Whichever code will be drawn, the player will win the 1st prizewhether the drawing concides with one of the 16 ticketspurchased, and 2nd prize in any other possible drawingcombination, as the "1-bit corrupted" combination willbe differing by one of the 16 played combinations !
In other words the set of 16 bit words guarantees that whateverthe drawing, there will be at least a second prize.
This simple example is particularly optimal and with 100%guarantee (there are only 128 different combinations of 7-bitbinary codes).
In the real lottery wheel case, there are millions of allpossible combinations, and an optimal set of codes or tickets toplay will hardly touch 100% chance.
There are many articles on Combinatorial Journals, where theratio "money won/money spent" varies a lot.
In the actual case of a state lottery the developed Hamming CodesTheory provides the HIGHEST POSSIBLE chance to match prizes !
NOTICE THAT a Hamming optimized system is NOT just going to be betterthan any other system, it is going to seize THE HIGHEST MATCHING CHANCE for ever in future humankind !