Redundancy as a purchase: what it buys, what it costs, and where the guarantee stops.
19 April 2027
Redundancy as a purchase: what it buys, what it costs, and where the guarantee stops.
Week 1 asked whether a copy changed at all, and answered it with a control that stayed identical. Week 7 modelled a process where change is not optional, only its rate is. This week asks the question those two leave open: if you know roughly what kind of damage to expect, what can you buy to survive it — and what does that purchase not cover?
A repetition code, stated completely#
Send payload bit 1 as 111. To recover it, take the majority of the three
received bits. That is the entire scheme: no arithmetic beyond counting, and
the workshop’s worked message — payload 1011, codeword 111 000 111 111 —
runs it end to end.
Detection is not correction. Two identical adjacent copies would let you notice a disagreement between them, but not say which one is right. A majority of three lets you notice and resolve a disagreement, provided at most one of the three is wrong. That “provided” is the entire content of the guarantee, and it is a guarantee about one bit-flip per triplet, stated in advance — not a general promise about noisy channels.
Where the guarantee stops#
Two flips inside the same triplet outvote the correct bit, and the decoder returns the wrong value with exactly the same confidence as a correct decode. The workshop’s three-case table makes this concrete: an unprotected bit that flips is simply wrong; a triplet with one flip self-corrects; a triplet with two flips also “self-corrects” — to the wrong answer, silently, having cost three times the bits to do it. Redundancy does not remove the risk of a wrong-but-confident answer. It relocates the number of simultaneous faults needed to produce one.
This is a cost-benefit purchase, not a law of nature: paying for a stronger code (more repeats, or a real error-correcting code rather than plain repetition) raises the number of simultaneous faults the scheme survives, at a higher overhead. The workshop keeps the arithmetic to plain repetition deliberately — a real correcting code is not required, because the shape of the tradeoff is the same one being taught.
Overhead is a number you calculate, not a slogan#
Overhead is the transmitted bits in excess of the payload, as a count and as a percentage of the payload. A 3x repetition code always has 200% overhead, regardless of message length, because every payload bit becomes three transmitted bits. That is a fixed price for a fixed guarantee (one flip per triplet) — the workshop’s costing exercise turns that price into a fictional dollar figure for a fictional collection, to make the tradeoff a decision rather than an abstraction.
Before Wednesday#
No pack to open. Read this page again and be ready to decode a received triplet by hand and to say, out loud, the one assumption that decode depends on.
Workshop: Copying without loss
Teaching team
- Dr Wren Halloway — convenor