Master Code forms Vector Space

Theorem

Let $\map V {n, p}$ be a master code of length $n$ modulo $p$.

Then $\map V {n, p}$ forms a vector space over $\Z_p$ of $n$ dimensions.

Proof

Recall the vector space axioms:

The vector space axioms consist of the abelian group axioms:

\((\text V 0)\)	$:$		Closure Axiom	\(\ds \forall \mathbf x, \mathbf y \in G:\)	\(\ds \mathbf x +_G \mathbf y \in G \)
\((\text V 1)\)	$:$		Commutativity Axiom	\(\ds \forall \mathbf x, \mathbf y \in G:\)	\(\ds \mathbf x +_G \mathbf y = \mathbf y +_G \mathbf x \)
\((\text V 2)\)	$:$		Associativity Axiom	\(\ds \forall \mathbf x, \mathbf y, \mathbf z \in G:\)	\(\ds \paren {\mathbf x +_G \mathbf y} +_G \mathbf z = \mathbf x +_G \paren {\mathbf y +_G \mathbf z} \)
\((\text V 3)\)	$:$		Identity Axiom	\(\ds \exists \mathbf 0 \in G: \forall \mathbf x \in G:\)	\(\ds \mathbf 0 +_G \mathbf x = \mathbf x = \mathbf x +_G \mathbf 0 \)
\((\text V 4)\)	$:$		Inverse Axiom	\(\ds \forall \mathbf x \in G: \exists \paren {-\mathbf x} \in G:\)	\(\ds \mathbf x +_G \paren {-\mathbf x} = \mathbf 0 \)

together with the properties of a unitary module:

\((\text V 5)\)	$:$		Distributivity over Scalar Addition	\(\ds \forall \lambda, \mu \in K: \forall \mathbf x \in G:\)	\(\ds \paren {\lambda + \mu} \circ \mathbf x = \lambda \circ \mathbf x +_G \mu \circ \mathbf x \)
\((\text V 6)\)	$:$		Distributivity over Vector Addition	\(\ds \forall \lambda \in K: \forall \mathbf x, \mathbf y \in G:\)	\(\ds \lambda \circ \paren {\mathbf x +_G \mathbf y} = \lambda \circ \mathbf x +_G \lambda \circ \mathbf y \)
\((\text V 7)\)	$:$		Associativity with Scalar Multiplication	\(\ds \forall \lambda, \mu \in K: \forall \mathbf x \in G:\)	\(\ds \lambda \circ \paren {\mu \circ \mathbf x} = \paren {\lambda \cdot \mu} \circ \mathbf x \)
\((\text V 8)\)	$:$		Identity for Scalar Multiplication	\(\ds \forall \mathbf x \in G:\)	\(\ds 1_K \circ \mathbf x = \mathbf x \)

First, the set of sequences $\tuple {x_1, x_2, \ldots, x_n}$, for $x_1, x_2, \ldots, x_n \in \Z_p$, has to be shown to fulfil the abelian group axioms.

This follows from:

Integers Modulo m under Addition form Cyclic Group

and:

Cyclic Group is Abelian.

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Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes