Our vectors are in extended Galois field (GF). We specifically have GF($2^4$) with primitive polynomial $\alpha^4 + \alpha + 1 = 0$ where $\alpha$ is the primitive element in extended Galois field GF($2^4$). The elements of the GF($2^4$) generated by $\alpha$ as such $\{0, 1, \alpha, \alpha^2, \alpha^3,\alpha^4,\alpha^5,\alpha^6,\alpha^7,\alpha^8,\alpha^9,\alpha^{10},\alpha^{11},\alpha^{12},\alpha^{13},\alpha^{14}\}$.
The question is the following how to create vectors that all the sub-set sums results in vectors that have specific property. Before going to that property, we first take one of the normal group under addition $\{0, \alpha, 1, \alpha^4\}$, call the elements to be $C_1$ type, with corresponding cosets $\{\alpha^2, \alpha^5, \alpha^8, \alpha^{10}\}$ call the elements $C_2$ types, $\{\alpha^{13}, \alpha^6, \alpha^{12}, \alpha^{11}\}$ call the elements $\{A, B, B, B\}$ type, and $\{\alpha^{14}, \alpha^3, \alpha^7, \alpha^{9}\}$ call the elements $\{B, A, A, A\}$ type. In addition let $0$ be $C'_1$ and $\alpha^2$ be $C'_2$. Therefore, property that need is that resultant vectors should have those rules
Any number of $C_1/C'_1$ (except for $C'_1$) and/or $C_2/C'_2$ (except for $C'_2$),
For each $C'_1$ there should be one $C'_2$ or vice versa
Else for each $C'_1$ there should be two $B$ types
Else for each $C'_2$ there should be two $A$ types
Else for each $A$ there should be one $B$ or vice versa.
The matrix below satisfy this condition. \begin{bmatrix} 0 & \alpha & 1 & \alpha^4 & 0 & \alpha & 1 & \alpha^4 \\ 0 & \alpha^6 & \alpha^{14} & \alpha^{11} & \alpha^6 & \alpha^8 & \alpha^{13} & \alpha^7 \\ 0 & \alpha^{13} & \alpha^9 & \alpha^7 & \alpha^{14} & \alpha & \alpha^6 & \alpha^{12} \end{bmatrix} Obviously, summation with first zero vector with any other vector is the vector itself e.g. let us give example of first with second vector, $[ 0, 0, 0 ]^T+ [ \alpha, \alpha^6, \alpha^{13}]^T = [ \alpha, \alpha^6, \alpha^{13}]^T$ and $[C_1, B, A]$ in terms of types. If we look rule 1. and 5. completely satisfies the property and this holds true with all additions of zero vector with any other vectors in the matrix. Another example add second with fifth vector, $[ \alpha, \alpha^6, \alpha^{13} ]^T+ [ 0, \alpha^6, \alpha^{14}]^T = [ \alpha, 0, \alpha^{2}]^T$ and $[C_1, C'_1, C'_2]$ in terms of types. The resulting vector satisfy the property via the rule 1. and 2. . So this vectors will satisfy the property for ALL the sub-set additions of the vectors.
The question is how to create those vectors in n dimension. Is there a specific structural to construct?