A good code exists and here is a good code are different sentences – some sixty years lay between them.
The corpus has carried the discipline of writing down which root an equals sign has. This paper applies that discipline to the form of a theorem itself: a theorem that asserts existence and a theorem that hands over an object have different roots. NO NEW MATHEMATICAL THEOREM IS CLAIMED. Shannon’s channel coding theorem, Reed-Solomon codes, Berlekamp-Welch decoding, the Singleton bound and the existence condition for finite fields are all standard.
First, Shannon says a code exists. The capacity of a binary symmetric channel is one minus the binary entropy, giving 0.919207 at a crossover of 0.01, then 0.713603, 0.500084 and 0.188722 at 0.05, 0.11 and 0.25 (computed here). But the spine of the proof is a bound on the error probability averaged over randomly drawn codes, and existence follows from one line: at least one must be as good as the average. No particular code appears anywhere in it. This does not mean the proof is incomplete – as an existence theorem it is complete. What differs is what one is handed.
Second, Galois hands over one. Over the field of eleven elements, a message of length three is treated as the coefficients of a polynomial and evaluated at eight distinct points, and a Berlekamp-Welch decoder was implemented alongside: a linear system in the error locator and its product with the message polynomial, solved and divided to recover the message. What is handed over is not only a code – a procedure for decoding comes with it. The existence theorem has none.
Third, and this is the core. The boundary is sharp. The message three, seven, two encodes to 1, 3, 9, 8, 0, 7, 7, 0. Introducing errors one at a time, zero, one and two errors are always repaired, and from three the system has no solution (computed here). The number of correctable errors is the floor of eight minus three over two, that is two, and the line falls exactly there. A construction theorem shows in numbers where the possible ends. The existence theorem draws no such line: it says that making the block long enough drives the error probability as low as one likes, and says nothing about how many errors this length-eight code repairs. An asymptotic statement and a finite one are different sentences.
Fourth, the construction is optimal as well. The Singleton bound says the minimum distance is at most eight minus three plus one, that is six. Enumerating all one thousand three hundred and thirty-one codewords, the Reed-Solomon minimum distance is exactly six. Drawing forty random linear codes of the same size, the minimum distance was four in twelve cases and five in twenty-eight, and not one of them reached six. The explicit construction sits on the bound every time and a code drawn at random usually does not: existing and being optimal are, again, two different questions. This paper does not write that random codes are bad – they are known to be good as the length grows, and Shannon’s proof rides on exactly that. What is shown is only that at this small size none of them reached the bound.
Fifth, the brake. The construction carries a constraint the existence theorem never had. A field of order q exists only when q is a prime power (Galois). Checking from two to thirty, fields exist for 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27 and 29, and there is none for 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28 and 30. Since there is no field of order six, there is no Reed-Solomon code over a six-letter alphabet either. Yet Shannon’s capacity is defined for any alphabet size whatever: only the construction bears the algebraic constraint. This does not make the construction inferior. It says only that the more concrete the thing handed over, the more concrete the constraints that come with it. A theorem that hands over nothing incurs no constraints on what it hands over.
The census. Shannon 1948 says a code exists, its tool is averaging over random codes, and its form is existence. Galois and Reed-Solomon say here is a code, the tool is polynomials over a finite field, and the form is construction, with an algorithm. Singleton says the minimum distance is at most n minus k plus one, the tool is counting, and the form is an upper bound. The existence condition for finite fields says only prime powers, the tool is Galois, and the form is a constraint. A good code exists and here is a good code are therefore not two phrasings of one fact: some sixty years lay between them, explicit constructions approaching capacity appearing more than half a century after Shannon’s theorem.
The corpus has met this shape repeatedly. In Paper 161, only one of five fixed-point theorems handed over an algorithm. In Paper 159, uniqueness was proved and recovery still was not assured. Writing down the form of a theorem’s conclusion is the same discipline as writing down the distinct root of a value.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none) / that random codes are bad (not claimed). What is added: building an encoder and a decoder here and showing the boundary falls exactly at t, comparing random linear codes against the bound numerically, giving the existence condition for finite fields its address as a constraint borne only by the construction, and setting existence and construction side by side in the corpus’s language of distinct roots. What remains: a proof of existence often hands over nothing at all. That is not a shortcoming; it is the kind of theorem it is. So the question to ask on reading one is not whether it was proved but what is left in one’s hands.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). “A Good Code Exists” and “Here Is a Good Code” Are Different Sentences [Paper 163]. Zenodo.
DOI: 10.5281/zenodo.22092677
ORCID: 0009-0006-8587-3450