More on Haucks warning

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Copyright Notice
The following manuscript
		EWD 528 More on Haucks warning :
is held in copyright by Springer-Verlag New York.
The manuscript was published as pages172173 of
	Edsger W. Dijkstra, Selected Writings on Computing: A Personal Perspective,
	Springer-Verlag, 1982. ISBN 0387906525.
Reproduced with permission from Springer-Verlag New York.
Any further reproduction is strictly prohibited.

More on Hauck’s warning.

In EWD525 “On a warning from E.A.Hauck” I mentioned without proof that with n=2^m bit there exist 2^n-m-1 different messages —I called them “codes”, but that is an unusual terminology for which I apologize— , such that any two different messages differ in at least four bit positions, thus allowing correction of one-bit errors and detection of two-bit errors. Since then I have been shown a proof of that theorem; I report that proof because it is so nice, and because it gives some further insights.

For the sake of brevity I shall demonstrate the theorem for 16:24 bits (in a way which is readily generalized for other values of m ). We consider 16 bits numbered from 0 through 15 , writing their index in binary:

d₀₀₀₀, d₀₀₀₁, d₀₀₁₀, d₀₀₁₁, ..., d₁₁₁₁

With “xxx1” we denote the set of odd indices, with “xx1x” the set {0010, 0011, 0110, 0111, 1010, 1011, 1110, 1111}, in general the set obtained by all possible substitutions of a 0 or a 1 at a place marked “x”, and define
h0 = parity(d_xxx1 ) , h1 = parity(d_xx1x ) , h2 = parity(d_x1xx ) , h3 = parity(d_1xxx )
where the function “parity” is = 0 if among the (8) bits with an index from the indicated set, the number of 1’s is even, and = 1 if it is odd. Further we introduce h = parity(d_xxxx) , which is just the sum of all the 16 bits modulo 2.

The 2¹¹ correct messages are then characterized by the equations

h0 = h1 = h2 = h3 = h = 0 .

Note. The above equations have indeed 2¹¹ different solutions: the 11 bits d₃, d₅, d₆, d₇, d₉, d₁₀, d₁₁, d₁₂, d₁₃, d₁₄, and d₁₅ can be chosen freely, we then solve h0 for d₁, h1 for d₂, h2 for d₄, and h3 for d₈, and finally h for d₀ .

We now denote by “a” the binary number formed by “h3 h2 h1 h0” and observe:
0)     for each correct message we have
          h = 0, a = 0
1)     for a one-bit error at bit position i we have
          h = 1, a = 1
2)     for a two-bit error at bit positions i and j
          h = 0, a = the bit-wise sum of i and j
(because i ≠ j , we conclude that a ≠ 0, thereby distinguishing this case from a correct message)
3)     for a three-bit error at positions i , j , and k
          h = 1, a = the bit-wise sum of i , j , and k.
4)     for a four-bit error at positions i , j , k , and l
          h = 0, a = the bit-wise sum of i , j , k , and l .

etc.

Under the assumption that one- and two-bit errors are the only errors that can occur, the rules are

h = 0 and a = 0:	accept the bit sequence
h = 1 :	invert bit d_a
h = 0 and a ≠ 0:	alarm, as two-bit error has been detected.

From the above, however, we see that all errors in 3, 5, 7, ... bits will then erroneously be interpreted as one-bit errors, i.e. in those cases our error correction indeed increases the probability of a wrong result being produced as if it were a correct one. The above gives a clear demonstration of the possible “harmfulness” of error correction alluded to in EWD525’s last paragraph. Hence this note.

Plataanstraat 5	prof.dr.Edsger W.Dijkstra
4565 - NUENEN	Burroughs Research Fellow
The Netherlands

Transcribed by Martin P.M. van der Burgt
Last revision 2014-11-27 .