![]() | 1 | initial version |
Since the ring is finite of order 33 and the matrix is 4×4, one approach is to try every possible vector; there are only (33)4=531441 possibilities.
![]() | 2 | No.2 Revision |
Since the ring is finite of order 33 and the matrix is 4×4, one approach is to try every possible vector; there are only (33)4=531441 possibilities.
A more refined approach is to introduce a vector with 3⋅4=12 undetermined coefficients, e.g. by making the base ring the quotient by t3 of the univariate polynomial ring in t over a polynomial ring in 12 variables over F3.
Considering the matrix over this base ring, you can multiply with this vector with undetermined coefficients, take the components, and set the coefficients of powers of t (of which the higher ones were automatically eliminated) equal to zero; this is an ordinary linear system over F3.
![]() | 3 | No.3 Revision |
Since the ring is finite of order 33 and the matrix is 4×4, one approach is to try every possible vector; there are only (33)4=531441 possibilities.
A more refined approach is to introduce a vector with 3⋅4=12 undetermined coefficients, e.g. by making the base ring the quotient by t3 of the univariate polynomial ring in t over a polynomial ring in 12 variables over F3.
Considering the matrix over this base ring, you can multiply with this vector with undetermined coefficients, take the components, and set the coefficients of powers of t (of which the higher ones were automatically eliminated) eliminated, due to the quotient by t3) equal to zero; this is an ordinary 12 times12 linear system over F3.
![]() | 4 | No.4 Revision |
Since the ring is finite of order 33 and the matrix is 4×4, one approach is to try every possible vector; there are only (33)4=531441 possibilities.
A more refined approach is to introduce a vector with 3⋅4=12 undetermined coefficients, e.g. by making the base ring the quotient by t3 of the univariate polynomial ring in t over a polynomial ring in 12 variables over F3.
Considering the matrix over this base ring, you can multiply with this vector with undetermined coefficients, take the components, and set the coefficients of powers of t (of which the higher ones were automatically eliminated, due to the quotient by t3) equal to zero; this is an ordinary $12 \ times \times 12linearsystemover\mathbb{F}_3$.