256 V Hadamard’s Determinant Problembe denoted mathematically by 0 or 1. On account of noise the message received may
differ slightly from that transmitted, and in some situations it is extremely important to
detect and correct the errors. One way of doing so would be to send the same message
many times, but it is an inefficient way. Instead suppose the message is composed of
codewordsof lengthn, taken from a subspaceUof the vector spaceV=Fn 2 .Thereare
2 kdifferent codewords, wherekis the dimension ofU. If the minimum weight of any
nonzero vector inUisd, then any two distinct codewords differ in at leastdplaces.
Hence if a codewordu∈Uis transmitted and the received vectorv∈Vcontains
less thand/2 errors, thenvwill be closer touthan to any other codeword. Thus if we
are confident that any transmitted codeword will contain less thand/2 errors, we can
correct them all by replacing each received vector by the codeword nearest to it.
The Golay code and the first-order Reed–Muller codes are of considerable practical
importance in this connection. For the first-order Reed–Muller codes there is a fast al-
gorithm for finding the nearest codeword toany received vector. Photographs of Mars
taken by the Mariner 9 spacecraft were transmitted to Earth, using the codeR( 1 , 5 ).
Othererror-correcting codesare used with compact discs to ensure high quality
sound reproduction by eliminating imperfections due, for example, to dust particles.8 FurtherRemarks
Kowalewski [22] gives a useful traditional account of determinants. Muir [28] is a
storehouse of information on special types of determinants; the early Japanese work is
described in Mikami [27].
Another approach to determinants, based on the work of Grassmann (1844), should
be mentioned here, as it provides easy access to their formal properties and is used in
the theory of differential forms. IfVis ann-dimensional vector space over a fieldF,
then there exists an associative algebraE,ofdimension2nas a vector space overF,
such that
(a)V⊆E,
(b)v^2 =0foreveryv∈V,
(c)VgeneratesE, i.e. each element ofEcan be expressed as a sum of a scalar mul-
tiple of the unit element 1 and of a finite number of products of elements ofV.
The associative algebraE, which is uniquely determined by these properties, is
called theGrassmann algebraorexterior algebraof the vector spaceV. It is easily
seen that any two products ofnelements ofVdiffer only by a scalar factor. Hence, for
any linear transformationA:V→V, there existsd(A)∈Fsuch that
(Av 1 )···(Avn)=d(A)v 1 ···vn for allv 1 ,...,vn∈V.Evidentlyd(AB)=d(A)d(B)and in factd(A)=detA, if we identifyAwith
its matrix with respect to some fixed basis ofV. This approach to determinants is
developed in Bourbaki [6]; see also Barnabeiet al.[4].
Dieudonn ́e (1943) has extended the notion of determinant to matrices with entries
from a division ring; see Artin [1] and Cohn [9]. For a very different method, see
Gelfand and Retakh [13].