The fullrank value function

Abstract

Motivated by applications to the wiretap channel of type II with the coset coding scheme as well as secret sharing, the concept of the fullrank value function is introduced in this paper. Several constructing methods for fullrank value functions are presented by using the finite projective geometry. The relationship between fullrank value functions and secret sharing schemes, and the relationship between fullrank value functions and separation of linear codes are given as well. In particular, new (2,2)-separating codes are constructed by using our methods.

Appendices

Appendix 1: Proof of Theorem 36

Proof

Assume $T=\{0, c_1\}$ and $T'=\{c_2, c_3\}$ are any two disjoint codewords subsets. Assume $c_1$, $c_2$, $c_3$, $c_1-c_2$ and $c_1-c_3$ correspond to $P_1$, $P_2$, $P_3$, $P_{12}$ and $P_{13}$ by using the correspondence given in Theorem 14, respectively. Then, $P_{12}$ contains $P_1\cap P_2$ and $P_{13}$ contains $P_1\cap P_3$. We may divide the analysis into several cases.

Consider $q\ge 3$ first.

(Case i) $c_1$, $c_2$ and $c_3$ are independent codewords. Then, $\dim (P_i\cap P_j)=k-3$ for $1\le i<j\le 3$ and $\dim (P_1\cap P_2\cap P_3)=k-4$. Note that the separating coordinate positions of T and $T'$ can be divided into two parts. One part is those positions at which $c_1$ is zero and both $c_2$ and $c_3$ are nonzero, and the other part is those positions at which all $c_i$, $1\le i\le 3$, are nonzero and at which both $c_1-c_2$ and $c_1-c_3$ are nonzero.

Observe that the number of the first part of separating coordinate positions is equal to m(S), where S is the set of points

$$\begin{aligned} S=\{p: m(p)>0 \text { and } p\in P_1 \text { and } p\notin (P_1\cap P_2) \text { and } p\notin (P_1\cap P_3)\}, \end{aligned}$$

and these points are located at $R\backslash (P_1\cap P_2\cap P_3)$ for all the $(k-3)$-dimensional subspaces R contained in $P_1$ and containing $P_1\cap P_2\cap P_3$, except $P_1\cap P_2$ and $P_1\cap P_3$. It can be checked that the number of the $(k-3)$-dimensional subspaces R above is $q-1$, and since $m(R\backslash (P_1\cap P_2\cap P_3))\ge 1$ for each R due to the fact that $m(\cdot )$ is $(k-3)$-fullrank, we have $m(S)\ge q-1$.

The number of the other part of separating coordinate positions is equal to $m(S')$, where $S'$ is the set of the points

$$\begin{aligned} \begin{aligned}&S'=\{p: p\in P, m(p)>0, p\notin (P\cap P_3), p\notin (P\cap P_{13}), P \text { is a } (k-2)\\&\quad -\text {dimensional subspace, } P\supset (P_1\cap P_2) \text { and } P\ne P_1, P_2, P_{12}\}. \end{aligned} \end{aligned}$$

It can be observed that the number of the $(k-2)$-dimensional subspaces P occurred in the set $S'$ is $q-2$, and for each such P, $P\cap S'$ consists of those points p which are located at those $R\backslash (P_1\cap P_2\cap P_3)$ and satisfies $m(p)>0$, where R is a $(k-3)$-dimensional subspace contained in P and containing $P_1\cap P_2\cap P_3$ and $R\ne P_1\cap P_2$, $P\cap P_3$ and $P\cap P_{13}$. It is obvious that the number of R above is $q-2$. Since $m(R\backslash (P_1\cap P_2\cap P_3))\ge 1$ due to the fact $m(\cdot )$ is $(k-3)$-fullrank, we get $m(P\cap S')\ge q-2$. Thus, $m(S')\ge (q-2)^2$.

Thus, the total number of separating coordinate positions of T and $T'$ is $m(S)+m(S')\ge q-1+(q-2)^2$.

(Case ii) The codewords $c_1$, $c_2$ and $c_3$ are dependent, but any two of them are independent. Then, $c_3$ is a linear combination of $c_1$ and $c_2$ with nonzero coefficients, and thus $P_1\cap P_2=P_1\cap P_2\cap P_3$. Similarly to the analysis of (Case i), the separating coordinate positions of T and $T'$ can be computed by $m(S)+m(S')$, where

$$\begin{aligned} S=\{p: p\in P_1, m(p)>0 \text { and } p\notin P_1\cap P_2\cap P_3\}, \end{aligned}$$

and

$$\begin{aligned} S'=\{p: m(p)>0, \text { and } p\notin (P_1\cup P_2\cup P_3\cup P_{12}\cup P_{13})\}. \end{aligned}$$

Thus, $m(S)+m(S')\ge q+q(q-4)\delta (q,4)$ due to the fact that $m(\cdot )$ is $(k-3)$-fullrank, where $\delta (q,4)$ is defined as

$$\begin{aligned} \delta (q,4)=\left\{ \begin{array}{ll} 1, &{} \quad q\ge 4;\\ 0, &{} \quad q<4. \end{array} \right. \end{aligned}$$

(Case iii)

If $c_1$, $c_2$ are independent, and $c_3$ and $c_1$ are dependent.

Then, the separating coordinate positions are those both $c_1$ and $c_2$ are nonzero and $c_1$ are not equal to $c_2$. The number of the separating coordinate positions is m(S), where

$$\begin{aligned} S=\{p: p\notin (P_1\cup P_2\cup P_{12})\} \end{aligned}$$

Thus, $m(S)\ge (q-2)q$ due to the fact that $m(\cdot )$ is $(k-3)$-fullrank.

Suppose that $c_1$, $c_2$ are independent, and $c_2$ and $c_3$ are dependent.

Then, the number of separating coordinate positions is $m(S)+m(S')$, where

$$\begin{aligned} S=\{p: m(p)>0, p\in P_1 \text { and } p\notin (P_1\cap P_2)\} \end{aligned}$$

and

$$\begin{aligned} S'=\{p: m(p)>0, p\notin (P_1\cup P_2\cup P_{12}\cup P_{13})\}. \end{aligned}$$

Since $m(\cdot )$ is $(k-3)$-fullrank, we have $m(S)\ge q$ and $m(S')\ge (q-3)q$, and thus $m(S)+m(S')\ge (q-2)q$.

Suppose that $c_2$, $c_3$ are independent, and $c_1$ and $c_2$ are dependent. Then, the number of separating coordinate positions is m(S), where

$$\begin{aligned} S=\{p: m(p)>0 \text { and } p\notin (P_1\cup P_3\cup P_{13})\}. \end{aligned}$$

Since $m(\cdot )$ is $(k-3)$-fullrank, we have $m(S)\ge (q-2)q$.

(Case iv) $c_1$ and $c_2$ are dependent, and $c_1$ and $c_3$ are also dependent. Then, the number of separating coordinate positions is m(S), where

$$\begin{aligned} S=\{p: p\notin P_1\}. \end{aligned}$$

Since $m(\cdot )$ is $(k-3)$-fullrank, we have $m(S)\ge q^2$.

Consider $q=2$.

Note that when $q=2$, only (Case i) and (Case ii) may occur. Summing up (Case i)–(Case iv), one may get the result of the theorem.$\square$

Appendix 2: Proof of Theorem 39

Proof

Using the same arguments as in Theorem 36, we compute the number of the separating coordinate positions case by case.

Consider $q\ge 3$ first.

(Case i) Since $m(\cdot )\equiv \nu$, we get $m(S)=q^{k-3}(q-1)m(\cdot )$ and $m(S')=q^{k-3}(q-2)^2m(\cdot )$ by using the same arguments as in (Case i) in Theorem 36. Thus, the total number of separating coordinate positions is $m(S)+m(S')=[q-1+(q-2)^2]q^{k-3}\nu$.

(Case ii) Similarly, using $m(\cdot )\equiv \nu$ and similar arguments as in (Case ii) in Theorem 36, we may obtain that $m(S)=q^{k-2}\nu$, and $m(S')$ $=$ $q^{k-2}(q-4)\delta (q,t)\nu$. Thus, $m(S)+m(S')$ $=$ $(q^{k-2}+q^{k-2}(q-4)\delta (q,4))\nu$.

(Case iii) Similar to the arguments as in (Case iii) in Theorem 36, we divide the analysis into several steps:

If $c_1$, $c_2$ are independent, and $c_3$ and $c_1$ are dependent, then $m(S)=(q-2)q^{k-2}\nu$.

If $c_1$, $c_2$ are independent, and $c_2$ and $c_3$ are dependent, then $m(S)+m(S')=(q-2)q^{k-2}\nu$.

If $c_2$, $c_3$ are independent, and $c_1$ and $c_2$ are dependent, then we have $m(S)=(q-2)q^{k-2}\nu$.

(Case iv) Using $m(\cdot )\equiv \nu$ and the same analysis as in (Case iv) in Theorem 36, we get $m(S)=q^{k-1}\nu$.

Thus,

$$\begin{aligned} \begin{aligned} \theta _{2,2}&= \min \{[q-1+(q-2)^2]q^{k-3}\nu , \ (q^{k-2}+q^{k-2}(q-4)\delta (q,4))\nu , \ \\&\quad \quad (q-2)q^{k-2}\nu , \ (q-2)q^{k-2}\nu , \ (q-2)q^{k-2}\nu , \ q^{k-1}\nu \}\\&=(1+(q-4)\delta (q,4))q^{k-2}\nu . \end{aligned} \end{aligned}$$

Consider $q=2$.

Then, only (Case i) and (Case ii) may occur. Using the same notation as in Theorem 36, in (Case i), the number of separating coordinate positions is equal to $m(S)=q^{k-3}(q-1)\nu =2^{k-3}\nu$, and in (Case ii), the number of separating coordinate positions is equal to $m(S)=q^{k-2}\nu =2^{k-2}\nu$.

Thus, $\theta _{2,2}=2^{k-3}\nu$.$\square$