Re: Complete solution to a modular System of equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg60475] Re: [mg60450] Complete solution to a modular System of equations*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sat, 17 Sep 2005 02:31:58 -0400 (EDT)*References*: <200509160750.DAA00817@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On 16 Sep 2005, at 16:50, mumat wrote: > In coding theory one is intersted to find the dual code of a given > code. To do this one has to solve a modular linear equation such as: > > > > eq = {a + b + d == 0, a + c + d == 0, b + c + d == 0} > > lhs = {{1, 1, 0, 1}, {1, 0, 1, 1}, {0, 1, 1, 1}}; rhs = {0, 0, 0}; > > LinearSolve[lhs,{0,0,0},Modulus->2] > > Out[52]= > {0,0,0,0} > > as you see it returns only one solution. How can I find all solutions. > The number of solutions is a power of 2. For about example there is > exacly one more solution which is > > {1,1,1,0}. > > Is there any function in Mathematica to do this? or I should start > writing my own code? > > thanks for your help in advance! > > regards, > > chekad > > eq = {a + b + d == 0, a + c + d == 0, b + c + d == 0}; Reduce[eq, {a, b, c, d}, Modulus -> 2] a == C[1] && b == C[1] && c == C[1] && d == 0 Here C[1] is an arbitrary constant in Z/2, in other words it can be either 0 or 1, hence the two solutions are exactly {0,0,0,0} and {1,1,1,0}. Andrzej Kozlowski

**References**:**Complete solution to a modular System of equations***From:*"mumat" <csarami@gmail.com>