Let \(\mathbf{1} \in \mathbb {R}^n\) be the all-ones vector. Then for every index \(i \in \{ 1, \dots , n\} \), the \(i\)-th coordinate satisfies \(\mathbf{1}_i = 1\).

Let \(x\) be a binary word of length \(n\), and let the \(\pm 1\) embedding send \(x\) to the vector in \(\mathbb {R}^n\) whose \(i\)-th coordinate is \(-1\) when \(x_i\) is true and \(+1\) when \(x_i\) is false. If the \(i\)-th entry of \(x\) is false, then the \(i\)-th coordinate of the embedded vector equals \(1\).

Let \(x\) be a binary word of length \(n\), and let \(s(x) \in \mathbb {R}^n\) denote its \(\pm 1\) embedding, whose \(i\)-th coordinate is \(-1\) when \(x_i\) is true and \(+1\) when \(x_i\) is false. Then for every index \(i\) with \(x_i\) true, the \(i\)-th coordinate of \(s(x)\) equals \(-1\).

Let \(x, y\) be binary words of length \(n\), and encode each as a \(\pm 1\) vector in \(\mathbb {R}^n\) by sending its \(i\)-th bit to \(-1\) when that bit is true and to \(+1\) when it is false. Then for every coordinate \(i\), the product of the \(i\)-th entries of these two embeddings equals \(1\) if \(x_i = y_i\) and \(-1\) if \(x_i \ne y_i\).

For any two binary words \(x, y\) of length \(n\), let \(\phi (x), \phi (y) \in \mathbb {R}^n\) be their \(\pm 1\) embeddings, whose \(i\)-th coordinates are \(-1\) when the corresponding bit is true and \(+1\) otherwise. Then their Euclidean inner product is

where \(d(x,y)\) is the Hamming distance, the number of coordinates on which \(x\) and \(y\) differ.

Let \(x\) be a binary word of length \(n\), and let \(s(x) \in \mathbb {R}^n\) denote its \(\pm 1\) embedding, the vector whose \(i\)-th coordinate is \(-1\) at each position where \(x_i\) is set and \(+1\) elsewhere. Writing \(\mathbf{1}\) for the all-ones vector and \(\mathrm{wt}(x)\) for the Hamming weight of \(x\) (the number of positions at which \(x\) is set), the \(\ell ^2\) inner product satisfies

Let \(\mathbf{1} \in \mathbb {R}^n\) denote the vector all of whose coordinates equal \(1\). Then, with respect to the standard \(\ell ^2\) inner product, \(\langle \mathbf{1}, \mathbf{1} \rangle = n\).

Let \(\alpha \in \mathbb {R}\), let \(x, y\) be binary words of length \(n\), and write \(\mathbf{1}\) for the all-ones vector in \(\mathbb {R}^n\) and \(s(x), s(y) \in \mathbb {R}^n\) for the \(\pm 1\) embeddings of \(x\) and \(y\) (with \(i\)-th coordinate \(-1\) where the corresponding bit is true and \(+1\) otherwise). Form the shifted vectors \(\hat{x}^\alpha = s(x) - \alpha \mathbf{1}\) and \(\hat{y}^\alpha = s(y) - \alpha \mathbf{1}\). Then their Euclidean inner product expands as

Let \(x\) and \(y\) be binary words of length \(n\), and let \(\alpha \ge 0\). Suppose \(d\) and \(w\) are natural numbers for which the Hamming distance between \(x\) and \(y\) is at least \(d\) and the Hamming weights of \(x\) and \(y\) are each at most \(w\). Writing \(\hat{x}^\alpha \) and \(\hat{y}^\alpha \) for the corresponding shifted vectors in \(\mathbb {R}^n\) — the \(\pm 1\) embedding of the word minus \(\alpha \) times the all-ones vector — their Euclidean inner product satisfies

For all natural numbers \(n\) and \(d\), the shift parameter \(\alpha (n,d) = \sqrt{(n - 2d)/n}\) satisfies \(0 \le \alpha (n,d)\).

Let \(n\) and \(d\) be natural numbers with \(n \gt 0\) and \(2d \le n\). Then the shift parameter \(\alpha (n,d)\) satisfies

Let \(n\) and \(d\) be natural numbers with \(n \ge 1\), \(d \ge 1\), and \(2d \le n\). Then the shift parameter satisfies

Let \(n\), \(d\), and \(w\) be natural numbers with \(n \gt 0\) and \(2d \le n\), and write \(\alpha = \alpha (n,d)\) for the shift parameter and \(J_2(n,d)\) for the binary Johnson radius. If \(w \le J_2(n,d)\), then

Let \(V\) be a real inner product space, let \(u \in V\) be a unit vector, and let \(x, y \in V\) satisfy \(\langle x, u\rangle \le 0\), \(\langle y, u\rangle \le 0\), and \(\langle x, y\rangle \le 0\). Then the components of \(x\) and \(y\) orthogonal to \(u\) again have nonpositive inner product:

Let \(V\) be a real inner product space, let \(u \in V\) be a unit vector, so \(\lVert u\rVert = 1\), and let \(x \in V\) be arbitrary. Then

Let \(V\) be a real inner product space, and let \(u, x \in V\) be unit vectors, \(\lVert u\rVert = 1\) and \(\lVert x\rVert = 1\), with \(\langle x, u\rangle \le 0\) and \(x \neq u\), \(x \neq -u\). Then the component of \(x\) orthogonal to \(u\) is nonzero, that is, \(x - \langle x, u\rangle \, u \neq 0\).

Let \(V\) be a real inner product space, and let \(u \in V\) be a unit vector. Let \(S \subseteq V\) be a set of unit vectors each of which makes a non-positive inner product with \(u\); that is, \(\lVert x\rVert = 1\) and \(\langle x, u\rangle \le 0\) for every \(x \in S\). Then the map \(x \mapsto x - \langle x, u\rangle \, u\), which subtracts from each vector its component along \(u\), is injective on \(S\).

Let \(V\) be a real inner product space, and let \(u,v \in V\) with \(\lVert u\rVert = 1\). Then the vector \(v - \langle u,v\rangle \, u\) lies in the orthogonal complement \((\mathrm{span}_{\mathbb {R}}\{ u\} )^{\perp }\) of the line spanned by \(u\).

Let \(u\) and \(v\) be unit vectors in a real inner product space, and suppose that \(v \neq u\) and \(v \neq -u\). Then the projection of \(v\) onto the orthogonal complement of \(\mathrm{span}\{ u\} \), namely \(v - \langle u,v\rangle \, u\), is nonzero.

Let \(V\) be a real inner product space, and let \(u,v,w \in V\) with \(\lVert u\rVert = 1\). Suppose the three pairwise inner products \(\langle v,w\rangle \), \(\langle v,u\rangle \), and \(\langle w,u\rangle \) are all nonpositive. For a vector \(x \in V\), write \(x - \langle u,x\rangle \, u\) for the projection of \(x\) onto the orthogonal complement of \(\mathrm{span}\{ u\} \). Then the projections of \(v\) and \(w\) again have nonpositive inner product:

Let \(u,v,w\) be unit vectors in a real inner product space satisfying \(\langle v,u\rangle \le 0\), \(\langle w,u\rangle \le 0\), and \(\langle v,w\rangle \le 0\), and suppose that neither \(v\) nor \(w\) equals \(u\) or \(-u\) and that \(v \neq w\). For \(x\) in the space write \(\pi (x) = x - \langle u,x\rangle \, u\) for the projection of \(x\) onto the orthogonal complement of \(\mathrm{span}\{ u\} \). Then the two normalized projections \(\lVert \pi (v)\rVert ^{-1}\, \pi (v)\) and \(\lVert \pi (w)\rVert ^{-1}\, \pi (w)\) cannot be equal.

Let \(u,v\) be unit vectors in a real inner product space, with \(v \neq u\) and \(v \neq -u\), and set \(w = v - \langle u,v\rangle \, u\), the projection of \(v\) onto the orthogonal complement of \(\mathrm{span}\{ u\} \). Then the rescaled vector \(\lVert w\rVert ^{-1}\, w\) has norm \(1\).

Let \(u,v,w\) be vectors in a real inner product space with \(\lVert u\rVert =1\), and suppose the three inner products \(\langle v,w\rangle \), \(\langle v,u\rangle \), and \(\langle w,u\rangle \) are all nonpositive. For a vector \(x\), write \(x^{\perp }=x-\langle u,x\rangle \, u\) for its component orthogonal to \(u\), and assume that both \(v^{\perp }\) and \(w^{\perp }\) are nonzero. Then their normalizations still have nonpositive inner product:

Let \(V\) be a finite-dimensional real inner product space, and let \(u \in V\) be a unit vector, so that \(\lVert u\rVert = 1\). Then the orthogonal complement of the line \(\mathrm{span}_{\mathbb {R}}\{ u\} \) has dimension one less than that of \(V\):

Let \(S\) be a finite subset of an additive group \(V\), and let \(u \in S\). Then

Let \(V\) be a real inner product space, let \(u \in V\) be a unit vector, and let \(v \in V\). Write \(w = v - \langle u, v\rangle \, u\) for the component of \(v\) orthogonal to \(u\). The normalized vector \(\lVert w\rVert ^{-1} w\), regarded as an element of the orthogonal complement \((\mathrm{span}_{\mathbb {R}}\{ u\} )^{\perp }\) of the line spanned by \(u\), has underlying vector in \(V\) equal to \(\lVert w\rVert ^{-1} w\).

Let \(n \ge 1\) and let \(\alpha \) be a real number with \(0 \le \alpha \lt 1\). Then for every binary word \(x\) of length \(n\), the shifted vector \(\hat{x}^\alpha \in \mathbb {R}^n\) — obtained from the \(\pm 1\) embedding of \(x\) by subtracting \(\alpha \) times the all-ones vector — is nonzero.

Fix natural numbers \(n\), \(d\), \(w\) and \(K\). If every \((n,d,w)\)-admissible code \(C\) satisfies \(\left\lvert C\right\rvert \le K\), then the maximum size of an \((n,d,w)\)-admissible code satisfies \(A_0(n,d,w) \le K\).