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4. d H (v, w) = d H (v + u, w + u) for all u, v, w ∈ V (translation invariance)
Definition 5.5.4
For λ ∈ N, a subset C ⊂ (F 2 ) n is called a λ–error correcting code, if
d H (u, v) ≥ 2λ + 1 for all u, v ∈ C with u ≠ v .
The reason for this name is the following
Lemma 5.5.5.
Let C ⊂ V be a λ-error correcting code. Then for any v ∈ V , there is at most one w ∈ C with
d H (v, w) ≤ λ.
Proof.
Suppose we have w 1 , w 2 ∈ C with d H (v, w i ) ≤ λ for i = 1, 2. Then the triangle inequality yields
d H (w 1 , w 2 ) ≤ d H (w 1 , v) + d H (v, w 2 ) ≤ 2λ .
Since the code C is supposed to be λ-error correcting, we have w 1 = w 2 .
✷
Remarks 5.5.6.
1. It is important to keep in mind the relative situation: a code C is a subspace of F n 2. The
Hemming distance gives an indication that it is important to keep the subspace C of code
words spread out in V .
2. We say that information is stored in the code, if an element c ∈ C is selected.
3. If C ⊂ F n 2, we say that a codeword of C is composed of n bits. If C is a linear code with
dim F2 C = k, we refer to a [n, k] code. Denote by
d :=
min d H (c, 0)
c∈C\{0}
the minimal distance of a code. We refer to an [n, k, d] code. In practice, the length n
of the code has to be kept small, because this causes costs for storing and transmitting.
The minimal distance d has to be big, since by lemma 5.5.5 this allows to many correct
errors. The dimension k of the code has to be big enough to allow enough code words.
From elementary linear algebra, one derives the singleton bound
k + d ≤ n + 1
which shows that these goals are in competition.
4. Classical storage devices are typically localized, either in space (e.g. an electron or a
nuclear spin) or in momentum space (e.g. a photon polarization).
5. Many storage devices are magnetic, i.e. a collection of coupled spins. The Hamiltonian
is such that it favours the alignment of spins. So if one spin is kicked out by thermal
fluctuation, the Hamiltonian tends to push it back in the right position. Thus errors in
the storage device are corrected by the dynamics of the system. This idea will also enter
in the construction of quantum codes.
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