TC+0 / 6. Hadamard codes S. Xambó Hadamard matrices. Paley’s construction of Hadamard matrices Hadamard codes. Decoding Hadamard codes Hadamard matrices A Hadamard matrix of order is a matrix of type whose coefficients are 1 or 1 and such that . Note that this relation is equivalent to say that the rows of have norm √ and that any two distinct rows are orthogonal. Since is invertible and we see that 2 and hence is also a Hadamard matrix. Thus is a Hadamard matrix if and only if its columns have norm √ and any two distinct columns are orthogonal. If is a Hadamard matrix of order , then det ⁄ . In particular we see that the Hadamard matrices satisfy the equality in Hadamard’s inequality: / |det| ∏∑ which is valid for all real matrices of order . Remark. The expression / is the supremum of the value of det when runs over the real matrices such that 1. Moreover, the equality is reached if and only if is a Hadamard matrix. 3 If we change the signs of a row (column) of a Hadamard matrix, the result is clearly a Hadamard matrix. If we permute the rows (columns) of a Hadamard matrix, the result is another Hadamard matrix. Two Hadamard matrices are called equivalent (or isomorphic) if it is possible to go from one to the other by means of a sequence of such operations. Observe that by changing the sign of all the columns that begin with 1 and afterwards of all the rows that begin with 1 we obtain an equivalent Hadamard matrix whose first row and first column only contain 1. Of such matrices we say that are normalized Hadamard matrices. 1 1 Example. is the unique normalized Hadamard matrix of 11 order 2. More generally if we define , 2, recursively by the formula 4 then is a normalized Hadamard matrix of order 2. It is also easy to check that if and are Hadamard matrices of orders and then the tensor product is a Hadamard matrix of order (the matrix also called the Kronecker product of and can be defined as the matrix obtained by substituting each component of by the matrix ). For example if is a Hadamard matrix of order then is a Hadamard matrix of order 2. Notice also that . 5 Proposition. If is a Hadamard matrix of order 3, then is divisible by 4. Proof. After permuting columns, the second row of a normalized Hadamard matrix of order may be assumed to have the form |. Since it has to be orthogonal to the first row, we get 0 or , and hence 2. After permuting columns again, we may assume that the third row (which exists if 3) has the form ||| as the number of 1’s and the number of 1 are both . Since this row is orthogonal to the second row, we get that 0 which implies that and hence 2, 4. 6 Remark. Any two Hadamard matrices of order 4 are equivalent. . The same happens for order 8 but later on we will see that this is no longer the case for order 12. 7 The Paley construction of Hadamard matrices Suppose that || is odd, so that with 2 prime. Let : 1 be the Legendre character of , namely 1 if is a square ⁄ 1 if not a square (conventionally we define 0 0). Note that ∑ 0. If we enumerate the elements of (in some order) 0, ,…, , the Paley matrix of , , is defined by the formula . In particular, the sum of the terms in any row (column) of is 0. 8 Proposition. if is the matrix with all its entries equal to 1, then has the following properties: 1) 2) 3) 4) 1/. Proof. 1) and 2) follow immediately from the fact that the sum of the elements of any row (column) of is 0. As for 3), the scalar product of a row of by itself has the form ∑ 1 and the scalar product by another has the form ∑ , 0. But ∑ ∑ 1 / 1 1. Finally / 1 1 . 9 Theorem. iF 10 mod4 1 then the matrix 1 is a Hadamard matrix of order . It is called the Hadamard matrix of . Proof. It is an easy corollary of the preceding proposition (1): 1 1 1 1 0 1. 0 1 Here are the 200 that satisfy this condition: {q in 5..200..2 such_that is_prime_power?(q) & rem(q+1, 4)==0} {7, 11, 19, 23, 27, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199}. 10 Remark. As we will see later on, in the case where 1 is not divisible by 4, it is possible to construct a Hadamard matrix of order 2 2 associated to . We are going to use the following concept: a matrix of order 1 is a conference matrix if its principal diagonal is zero, the other elements are 1 and 1. Remarks a) Two distinct rows of a conference matrix are orthogonal and the norm of each row is √1. A similar statement is valid for columns. i) If we change the sign of a row (column) of , the result is also a conference matrix. ii) If we transpose two rows of and then the two columns with the same indices, the result is also a conference matrix. 11 Two conference matrices are said to be equivalent if we can transform one into the other using operations of type i and ii. For example, any conference matrix is equivalent to a normalized conference matrix, which means that the first row and column are equal to 0. Moreover, we can also make sure that the second row has the form 0 . Since this has to be orthogonal to the first row, we conclude that the order of a conference matrix is even. b) Let be the Paley matrix of a finite field , odd, and let 1 if is symmetric and 1 if is skew‐symmetric ( ). Then 0 is a conference matrix of order 1 (the conference matrix of ). The matrix is symmetric if 1 mod 4 and skew‐symmetric if 3 mod 4. 12 Remark. a) If is an skew‐symmetric conference matrix of order , is a Hadamard matrix: . Note that if is the conference matrix of Remark b) above, then is 1 equivalent to , which is the matrix considered in the theorem (p. 9). b) If is a symmetric conference matrix of order is a Hadamard matrix of order 2 (a similar calculation). c) If is the conference matrix of a finite field and 1 is not divisible by 4, then b) yields a Hadamard matrix of order 2 2. This matrix will be called the Hadamard matrix de . 13 Remark. The Hadamard matrices , , , have order 4, 8, 16, 32. For 7, 11, 19, 23, 27, 31, the theorem (p. 9) yields Hadamard matrices , , , , , . Part c) of the preceding Remark applied to the fields , and , provides us with matrices , , , . Thus we have Hadamard matrices of orders 4, 8, 12, 16, 20, 24, 28, 32 and 36. Proceeding in similar ways and with others that are beyond the scope of these notes,N1 we can construct Hadamard matrices for many orders. The first order for which no Hadamard matrix is known is 668 (the previous value was 428, but it was solved by Behruz Tayfeh‐ Rezaie in 2004).N2 It is conjectured that for any that is multiple of 4 there is a Hadamard matrix of order . This is usually called the Hadamard conjecture, but it is probably due to Raymond Paley. 14 Remark. The matrices and are distinct, but they are equivalent (all the Hadamard matrices of order 8 are equivalent). It can also be seen that the matrices and are equivalent, even though it turns out that there are Hadamard matrices of order 12 that are not equivalent. 15 Code associated to a Hadamard matrix We associate a code of type , 2 to any Hadamard matrix of order as follows: where () is the binary matrix obtained by the substitution 1 0 in (). Notice that the words in can also be obtain as the complements of the words in . Since two rows of differ in exactly /2 positions, the minimum distance of is /2. Hence is a code of type , 2, /2. In general, this code is not linear (a necessary condition for linearity is that be a power of 2). Note that is not equidistant, as it contains the words and . 16 Example. The code is equivalent to the parity completion of the Hamming code 7, 4, 3. In particular it is a linear code. 1 1 1 1 1 1 1 1 1234 1 0 1 1 0 1 0 0 134 1 0 0 1 1 0 1 0 14 1 0 0 0 1 1 0 1 1 0 1 1 0 0 1 1 0 12 1 0 1 0 0 0 1 1 13 1 1 0 1 0 0 0 1 124 1 1 1 0 1 0 0 0 123 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 2 0 1 1 0 0 1 0 1 23 0 1 1 1 0 0 1 0 234 0 0 1 1 1 0 0 1 34 0 1 0 1 1 1 0 0 24 0 0 1 0 1 1 1 0 3 0 0 0 1 0 1 1 1 4 17 Example. The code 32, 64, 16 is the code used in the period 1969‐1972 by the Mariner spacecraft to transmit images of Mars (this code also turns out to be linear and actually equivalent to the (linear) code 5).