By P. P. Martin
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Extra resources for Bottom-up Computing and Discrete Mathematics
We assume that this sequence has value to us for some reason (determining the humanistic value of a given sequence is beyond the scope of this section, but it might contain, for example, a list of transactions in your bank account for the last year). The challenge we face is to transmit this data to a new location. For example, perhaps a person in England wants to communicate the question implied by the sequence (w, i, l, l, y, o, u, m, a, r, r, y, m, e) to a friend in Australia. In this case obviously shouting it out, or writing it onto a sheet of paper and throwing this in a southerly direction, is not going to get the job done, even if the recipient knows to expect a message (audible or written, respectively) in some given time-window.
1): (a) If d(C) ≥ t + 1 then no x lies in another’s ball. Thus if 1 up to t errors occur then the received message is not in C and we know we have an error. (b) If d(C) ≥ 2t + 1 then even the balls are disjoint (this is perhaps not so obvious with the Hamming distance, cf. say the usual Euclidean metric, but the triangle inequality is what we need to confirm it), and if 1 up to t errors occur then the received message is closer to x than any other y ∈ C. 1 Optimisation Code choice affects transmission error probability We are making a case, superficially, that f3 is more reliable than f1 when transmitting over a channel with errors.
Encoded version travels somehow to B, degrading gradually for various 6 On the other hand there is no system which will enable us to encode even a single ‘generic’ element of the set (0, 1) (the unit open real interval). Some elements in this interval can be communicated by more abstract means. For √ example π, e and 2. Such abstractions are presently among the features distinguishing humans from computers... but that is another story. 2. CODING reasons as it travels; 4. Degraded encoded version reaches target’s decoder (nominally in our example it is a sound, so the decoder is an ear/brain system; but obviously the sound heard by B at the appropriate point in time will have only a negligible amount of correlation with the original encoding).