Let \(p\) be prime and work over the space \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) equipped with the symplectic form \(\omega (u,v) = \sum _{i=0}^{n-1}(x_i z'_i - z_i x'_i)\), where \(u = (x,z)\) and \(v = (x',z')\). Then for all \(x, y, z \in V\),

Fix a prime \(p\) and a natural number \(n\), and let \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\), where \(\mathbb {F}_p = \mathbb {Z}/p\mathbb {Z}\). Equip \(V\) with the symplectic form \(\omega (u,v) = \sum _{i=0}^{n-1}(x_i z'_i - z_i x'_i)\) for \(u = (x,z)\) and \(v = (x',z')\). Then \(\omega \) is additive in its second argument: for all \(x, y, z \in V\),

Let \(p\) be prime and work over the prime field \(\mathbb {F}_p\), with \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) carrying the symplectic form \(\omega \) given by \(\omega (u,v) = \sum _{i=0}^{n-1}(x_i z'_i - z_i x'_i)\) for \(u=(x,z)\) and \(v=(x',z')\). Then for every scalar \(c \in \mathbb {F}_p\) and all \(x, y \in V\),

where \(c\, x\) denotes the coordinatewise scalar multiplication of \(x\) by \(c\).

Let \(p\) be a prime, and equip \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) with the symplectic form \(\omega (u,v) = \sum _{i=0}^{n-1}(x_i z'_i - z_i x'_i)\), where \(u = (x,z)\) and \(v = (x',z')\). Then for every scalar \(c \in \mathbb {F}_p\) and all \(x, y \in V\),

where \(c\, y\) denotes componentwise scalar multiplication.

Fix a prime \(p\) and an integer \(n\), and equip \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) with the bilinear form \(\omega (u,v) = \sum _{i=0}^{n-1}(x_i z'_i - z_i x'_i)\), where \(u = (x,z)\) and \(v = (x',z')\). Then \(\omega \) is antisymmetric: for all \(u, v \in V\),

Let \(p\) be a prime and \(n \ge 0\), and let \(\omega \) be the symplectic form on \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\). If a vector \(u \in V\) satisfies \(\omega (u,v) = 0\) for every \(v \in V\), then \(u = 0\).

Fix a prime \(p\) and a natural number \(n\), and let \(v \in \mathbb {F}_p^n \times \mathbb {F}_p^n\). Then the weight \(\mathrm{wt}(v)\) satisfies \(\mathrm{wt}(v) \le n\).

Fix a prime \(p\) and an integer \(n\), and let \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\) be the space of pairs of coordinate vectors over the prime field \(\mathbb {F}_p\). For a subset \(C \subseteq \{ 1, \dots , n\} \), let \(V_C \subseteq V\) be the support submodule consisting of those pairs \((u, w)\) for which \(u_i = 0\) and \(w_i = 0\) at every index \(i \notin C\). Then \(V_C\) has dimension \(\dim _{\mathbb {F}_p}(V_C) = 2\lvert C\rvert \) over \(\mathbb {F}_p\).

Let \(p\) be a prime and equip the \(2n\)-dimensional space \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\) over the prime field \(\mathbb {F}_p\) with the symplectic form \(\omega \big((x,z),(x',z')\big) = \sum _{i=1}^{n}(x_i z'_i - z_i x'_i)\). Then for every subspace \(S \le V\), its symplectic orthogonal complement \(S^{\perp _\omega } = \{ v \in V \mid \omega (v,s) = 0 \text{ for all } s \in S\} \) has dimension

Fix a prime \(p\) and a natural number \(n\), and let \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\) carry the symplectic form \(\omega \bigl((x,z),(x',z')\bigr) = \sum _{i} (x_i z'_i - z_i x'_i)\). For every \(\mathbb {F}_p\)-subspace \(S \le V\), the associated code distance \(d(S)\) — the least weight \(\mathrm{wt}(v) = |\mathrm{supp}(v)|\) attained by a vector \(v\) lying in the symplectic orthogonal complement \(S^{\perp _\omega }\) but not in \(S\) — satisfies \(d(S) \le n\).

Let \(p\) be prime and let \(S\) be an \(\mathbb {F}_p\)-subspace of the symplectic space \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\), with code distance \(d(S)\) and symplectic orthogonal complement \(S^{\perp _\omega }\). If \(E \subseteq \{ 0,\dots ,n-1\} \) is a set of erased coordinates with \(|E| \lt d(S)\), then \(E\) is correctable for \(S\): every \(v \in V_E \cap S^{\perp _\omega }\) already lies in \(S\).

Work in the symplectic space \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\) over the prime field \(\mathbb {F}_p\), equipped with the symplectic form \(\omega \), and let \(S \le V\) be an isotropic submodule, so that \(\omega (u,v) = 0\) for all \(u, v \in S\). Suppose \(A, B \subseteq \{ 0, 1, \dots , n-1\} \) are two disjoint sets of coordinates, each correctable for \(S\): every vector of \(S^{\perp _\omega }\) supported on \(A\) lies in \(S\), and likewise every vector of \(S^{\perp _\omega }\) supported on \(B\) lies in \(S\). Then the logical dimension \(k(S) = n - \dim S\) satisfies

Fix a prime \(p\) and an integer \(n\), and set \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\), equipped with the alternating bilinear form

Let \(B \colon V \times V \to \mathbb {F}_p\) be the bilinear form on \(V\) obtained by bundling \(\omega \). Then \(B\) agrees with \(\omega \) on every pair of arguments: for all \(u, v \in V\) one has \(B(u,v) = \omega (u,v)\).

Fix a prime \(p\), a natural number \(n\), and a subset \(C \subseteq \{ 0,1,\dots ,n-1\} \) of coordinates, and let \(V_C\) denote the support submodule of \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) consisting of vectors whose support lies in \(C\). Let \(E \colon (C \to \mathbb {F}_p) \times (C \to \mathbb {F}_p) \to V_C\) be the extension map, which sends a pair \((f,g)\) to the vector agreeing with \(f\) and \(g\) on \(C\) and equal to zero off \(C\), and let \(R \colon V_C \to (C \to \mathbb {F}_p) \times (C \to \mathbb {F}_p)\) be the restriction map, which sends a vector supported on \(C\) to its two component functions restricted to \(C\). Then \(R \circ E\) is the identity: for every pair \(x = (f,g)\) of functions \(f, g \colon C \to \mathbb {F}_p\) one has \(R(E(x)) = x\).

Fix a prime \(p\) and a subset \(C \subseteq \{ 1,\dots ,n\} \) of coordinates, and let \(V_C \subseteq \mathbb {F}_p^n \times \mathbb {F}_p^n\) be the submodule of pairs of vectors whose support is contained in \(C\). Then extending back the coordinate functions of a supported vector recovers that vector exactly: for every \(x \in V_C\), applying the restriction map \(V_C \to (C \to \mathbb {F}_p) \times (C \to \mathbb {F}_p)\) and then the extension-by-zero map back into \(V_C\) returns \(x\). Equivalently, the extension map is a left inverse of the restriction map.

Fix a prime \(p\) and let \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\). For a subset \(E \subseteq \{ 1,\dots ,n\} \), let \(r_E \colon V \to V_E\) be the restriction map that keeps the coordinates indexed by \(E\) and sets all others to zero. Then the kernel of \(r_E\) is the support submodule \(V_{E^c}\) consisting of those vectors whose support lies in the complement \(E^c = \{ 1,\dots ,n\} \setminus E\); that is, \(r_E\) annihilates exactly the vectors that vanish on every coordinate in \(E\).

Let \(E \subseteq \mathrm{Fin}\, n\) be a finite subset. Then its complement equals the set difference of the whole space and \(E\); that is, \(E^c = \mathrm{Fin}\, n \setminus E\).

Fix a prime \(p\) and an integer \(n \ge 0\), and let \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\). For a set \(E\) of coordinates, let \(r_E \colon V \to V\) be the restriction map that leaves each coordinate in \(E\) unchanged and sets every coordinate outside \(E\) to zero, and for a set \(C\) of coordinates let \(V_C \le V\) be the subspace of vectors supported on \(C\). Then for every subspace \(S \le V\),

where \(E^c\) denotes the complement of \(E\) among the \(n\) coordinates.

Fix a prime \(p\) and an integer \(n\), and work in the space \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\) equipped with the symplectic form \(\omega \). Let \(M \subseteq \{ 0, 1, \dots , n-1\} \) be a set of coordinates, and let \(v \in V_M\), so that every coordinate of \(v\) lying outside \(M\) vanishes. Then for every \(s \in V\),

where \(r_M(s)\) is the vector obtained from \(s\) by zeroing out all coordinates outside \(M\).

Let \(p\) be prime, \(\mathbb {F}_p\) the field with \(p\) elements, and let \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) carry the symplectic form \(\omega \big((x,z),(x',z')\big) = \sum _{i=0}^{n-1}(x_i z'_i - z_i x'_i)\). For a subset \(M \subseteq \{ 0,\dots ,n-1\} \), let \(r_M : V \to V\) be the endomorphism that keeps the coordinates indexed by \(M\) and sets all others to zero, and let \(V_M \subseteq V\) be the subspace of vectors whose support is contained in \(M\). Then for every \(s \in V\) and every \(v \in V_M\),

Let \(p\) be a prime, let \(n\) be a natural number, and let \(M \subseteq \{ 0,1,\dots ,n-1\} \) be a subset of coordinates. Write \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) for the symplectic space equipped with the form \(\omega \big((x,z),(x',z')\big) = \sum _{i=0}^{n-1}(x_i z'_i - z_i x'_i)\), and let \(V_M \subseteq V\) be the support submodule of those vectors whose coordinates outside \(M\) all vanish. If \(v \in V_M\) satisfies \(\omega (v,w) = 0\) for every \(w \in V_M\), then \(v = 0\).

Fix a prime \(p\) and a natural number \(n\), and let \(M \subseteq \{ 0,1,\dots ,n-1\} \) be a set of coordinates. For every \(s \in V\) and every \(v \in V_M\) (that is, every \(v\) supported on \(M\)),

In words: applying the restriction endomorphism \(r_M^V\), which zeroes out the coordinates outside \(M\), to the left argument of the symplectic form leaves the value unchanged whenever the right argument is supported on \(M\).

Fix a prime \(p\) and work in the symplectic space \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\) equipped with the form \(\omega \big((x,z),(x',z')\big) = \sum _{i=0}^{n-1}(x_i z'_i - z_i x'_i)\), and for a subset \(M \subseteq \{ 0,\dots ,n-1\} \) let \(V_M\) be the submodule of vectors whose support lies in \(M\) and let \(r_M \colon V \to V\) be the map that zeroes every coordinate outside \(M\) and fixes those in \(M\). Then for every \(\mathbb {F}_p\)-subspace \(S \le V\),

where \(r_M(S)\) denotes the image of \(S\) under \(r_M\) and \((\cdot )^{\perp _\omega }\) the orthogonal complement with respect to \(\omega \).

Fix a prime \(p\) and let \(\omega \) denote the symplectic form on \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\). For a subset \(M \subseteq \{ 0, \dots , n-1\} \) and any two vectors \(x, y\) in the support submodule \(V_M\), the restricted symplectic form evaluated at \((x, y)\) agrees with \(\omega (x, y)\) computed on \(x\) and \(y\) regarded as elements of the ambient space \(V\).

Let \(p\) be a prime, and equip \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) with the symplectic form \(\omega \big((x,z),(x',z')\big) = \sum _{i=0}^{n-1}(x_i z'_i - z_i x'_i)\). For any subset \(M \subseteq \{ 0,\dots ,n-1\} \), let \(V_M\) be the support submodule consisting of those \((x,z) \in V\) with \(x_i = z_i = 0\) for every \(i \notin M\). Then the restriction of \(\omega \) to \(V_M\) is nondegenerate: if \(u \in V_M\) satisfies \(\omega (u,v) = 0\) for all \(v \in V_M\), then \(u = 0\).

Let \(p\) be a prime and let \(\omega \) be the symplectic form on \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\). For any subset \(M\) of \(\{ 0, 1, \dots , n-1\} \) and any submodule \(S \le V\), the vectors of the support submodule \(V_M\) that are \(\omega \)-orthogonal to every element of \(S\) are exactly the images, under the inclusion \(V_M \hookrightarrow V\), of the vectors of \(V_M\) that are orthogonal to \(r_M(S)\) with respect to the restriction of \(\omega \) to \(V_M\). In symbols, writing \(\iota \colon V_M \hookrightarrow V\) for the inclusion and \(\perp _M\) for orthogonality with respect to the restricted form,

where \(r_M(S)\) is the image of \(S\) under the restriction map \(r_M \colon V \to V_M\).

Fix a prime \(p\), a natural number \(n\), and a subset \(M \subseteq \{ 0,1,\dots ,n-1\} \), and let \(V_M\) be the associated support submodule of \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\). Then the restriction to \(V_M\) of the symplectic form \(\omega (u,v) = \sum _{i} (x_i z'_i - z_i x'_i)\), where \(u=(x,z)\) and \(v=(x',z')\), is reflexive: for all \(u, v \in V_M\), if \(\omega (u,v) = 0\) then \(\omega (v,u) = 0\).

Fix a prime \(p\), an integer \(n\), and work in the symplectic space \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\) equipped with the symplectic form \(\omega \big((x,z),(x',z')\big) = \sum _{i} (x_i z'_i - z_i x'_i)\). Let \(M\) be a subset of the coordinate set \(\{ 0,1,\dots ,n-1\} \), let \(V_M \le V\) be the support submodule of vectors whose coordinates vanish outside \(M\), and let \(r_M : V \to V_M\) be the restriction map that zeroes every coordinate outside \(M\). Then for any \(\mathbb {F}_p\)-subspace \(S \le V\), the symplectic orthogonal complement \(S^{\perp _\omega }\) satisfies

Fix a prime \(p\) and equip \(V = \mathbb {F}_p^{\, n} \times \mathbb {F}_p^{\, n}\) with the symplectic form \(\omega \big((x,z),(x',z')\big) = \sum _{i=0}^{n-1}\big(x_i z'_i - z_i x'_i\big)\). For a set of coordinates \(M \subseteq \{ 0,\dots ,n-1\} \), let \(V_M \le V\) be the subspace of vectors supported on \(M\), and for a subspace \(W \le V\) write \(W_M = W \cap V_M\). Let \(S \le V\) be isotropic, meaning \(\omega (u,v) = 0\) for all \(u,v \in S\), let \(S^{\perp }\) denote its symplectic orthogonal complement, and set \(g(S,M) = \dim _{\mathbb {F}_p}\! \big(S^{\perp }\cap V_M\big) - \dim _{\mathbb {F}_p}(S_M)\). Then, writing \(M^c\) for the complementary set of coordinates,

Fix a natural number \(n\), and for a subset \(E \subseteq \mathrm{Fin}\, n\) write \(E^c = \mathrm{Fin}\, n \setminus E\) for its complement. Then for every subset \(M \subseteq \mathrm{Fin}\, n\), taking the complement twice recovers the original set: \((M^c)^c = M\).

Fix a prime \(p\) and an integer \(n\), and let \(S\) be a subspace of the ambient space \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\). For a subset \(M \subseteq \{ 1,\dots ,n\} \) with complement \(M^c\), write \(S_M = S \cap V_M\) and \(S_{M^c} = S \cap V_{M^c}\) for the intersections of \(S\) with the corresponding support subspaces. Then

Let \(p\) be prime, and equip the symplectic space \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) with its standard alternating form \(\omega \). If \(S \le V\) is an isotropic subspace and \(M \subseteq \{ 0,1,\dots ,n-1\} \) is any set of coordinates, then

where \(M^c\) is the complement of \(M\) and \(S_{M^c} = S \cap V_{M^c}\) consists of the vectors of \(S\) supported on \(M^c\).

Let \(S\) be an isotropic submodule of the symplectic space \(V=\mathbb {F}_p^n\times \mathbb {F}_p^n\), and let \(M\subseteq \{ 0,\dots ,n-1\} \) be a set of coordinates with complement \(M^c\). Writing \(S_M=S\cap V_M\) and \(S_{M^c}=S\cap V_{M^c}\) for the parts of \(S\) supported on \(M\) and on its complement, one has

Let \(p\) be a prime and \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\), equipped with the symplectic form \(\omega \big((x,z),(x',z')\big) = \sum _{i} (x_i z'_i - z_i x'_i)\). For a coordinate subset \(M \subseteq \{ 0,\dots ,n-1\} \), let \(V_M \le V\) be the support submodule of vectors supported on \(M\), and for a subspace \(S \le V\) write \(S_M = S \cap V_M\). If \(S\) is isotropic, that is \(\omega (u,v) = 0\) for all \(u,v \in S\), then

where \(M^c = \{ 0,\dots ,n-1\} \setminus M\) is the complementary coordinate set.

Let \(p\) be prime and let \(S \le V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) be a subspace that is isotropic for the symplectic form \(\omega \), meaning \(\omega (u,v)=0\) for all \(u,v \in S\). Then for every subset \(M \subseteq \{ 0,\dots ,n-1\} \), with complement \(M^c\),

where \(g(S,M) = \dim _{\mathbb {F}_p}\! \big(S^{\perp }_M\big) - \dim _{\mathbb {F}_p}\! \big(S_M\big)\).

Let \(n\) be a natural number and let \(M\) be a subset of the \(n\)-element index set \(\mathrm{Fin}\, n = \{ 0, 1, \dots , n-1\} \). Then the size of \(M\) and the size of its complement \(M^c = \mathrm{Fin}\, n \setminus M\) add up to \(n\):

Fix a prime \(p\), let \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) be equipped with the symplectic form \(\omega \big((x,z),(x',z')\big) = \sum _{i} (x_i z'_i - z_i x'_i)\), and let \(S \le V\) be a subspace with symplectic orthogonal complement \(S^{\perp _\omega } = \{ v \in V : \omega (v,s) = 0 \text{ for all } s \in S\} \). Let \(M \subseteq \{ 0,1,\dots ,n-1\} \) be a set of coordinates, write \(V_M\) for the subspace of vectors supported on \(M\), and set \(S_M = S \cap V_M\) and \(S^{\perp }_M = S^{\perp _\omega } \cap V_M\). If \(M\) is correctable for \(S\) — meaning every vector of \(S^{\perp _\omega }\) supported on \(M\) already lies in \(S\), i.e. \(S^{\perp }_M \le S\) — then \(g(S,M) = \dim _{\mathbb {F}_p}(S^{\perp }_M) - \dim _{\mathbb {F}_p}(S_M) = 0\).

Let \(S\) be an isotropic subspace of the symplectic space \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) (with \(p\) prime), and let \(M \subseteq \{ 0,1,\dots ,n-1\} \) be an erasure set that is correctable for \(S\). Writing \(M^c\) for the complement of \(M\), one has

Let \(p\) be prime and let \(S\) be an \(\mathbb {F}_p\)-subspace of the symplectic space \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\). Suppose \(B\) and \(C\) are disjoint subsets of the coordinate set \(\{ 0,\dots ,n-1\} \), and that \(B\) is correctable for \(S\). Then

where \(g(S,M) = \dim _{\mathbb {F}_p}\! \bigl(S^{\perp }_M\bigr) - \dim _{\mathbb {F}_p}\! \bigl(S_M\bigr)\).

Fix a prime \(p\) and an integer \(n\), and let \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\). Taking \(C\) to be the full index set \(\{ 0,1,\dots ,n-1\} \), the support submodule \(V_C = \{ v \in V \mid \mathrm{supp}(v) \subseteq C\} \) is all of \(V\); that is, \(V_C = V\).

Let \(p\) be a prime and \(\mathbb {F}_p\) the field with \(p\) elements, and for a natural number \(n\) set \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\). Then \(V\) is a vector space over \(\mathbb {F}_p\) of dimension

Fix a prime \(p\) and an integer \(n\), and let \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) carry the bilinear form

over the prime field \(\mathbb {F}_p\). Then \(\omega \) is non-degenerate: the only vector \(u \in V\) with \(\omega (u,v) = 0\) for every \(v \in V\) is \(u = 0\).

Let \(p\) be a prime, \(n\) a natural number, and let \(\omega \) be the bilinear form on \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) given by \(\omega (u,v) = \sum _{i=1}^{n}(x_i z'_i - z_i x'_i)\) for \(u = (x,z)\) and \(v = (x',z')\). Then \(\omega \) is reflexive: for all \(u, v \in V\), if \(\omega (u,v) = 0\) then \(\omega (v,u) = 0\).

Let \(p\) be a prime and equip the \(2n\)-dimensional space \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\) over \(\mathbb {F}_p\) with the symplectic form \(\omega \big((x,z),(x',z')\big) = \sum _{i=1}^{n} (x_i z'_i - z_i x'_i)\). If \(S \le V\) is an isotropic subspace, meaning \(\omega (u,v) = 0\) for all \(u, v \in S\), then \(\dim _{\mathbb {F}_p} S \le n\).

Let \(p\) be a prime and let \(S\) be a subspace of the symplectic space \(V = \mathbb {F}_p^n \times \mathbb {F}_p^n\). If \(S\) is isotropic and its logical dimension vanishes, \(k(S) = 0\), then the code distance vanishes as well, \(d(S) = 0\).

Let \(n\) and \(t\) be natural numbers with \(2t \le n\). Then there exist two disjoint subsets \(A, B \subseteq \{ 1, \dots , n\} \), each of cardinality \(t\).