Re: Complete solution to a modular System of equations

• To: mathgroup at smc.vnet.net
• Subject: [mg60466] Re: [mg60450] Complete solution to a modular System of equations
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Sat, 17 Sep 2005 02:31:47 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```eq={a+b+d==0,a+c+d==0,b+c+d==0};

soln=Reduce[eq,{a,b,c,d},Modulus->2]

a == C[1] && b == C[1] && c == C[1] && d == 0

List@@Last/@soln/.{{C[1]->0},{C[1]->1}}

{{0, 0, 0, 0}, {1, 1, 1, 0}}

Bob Hanlon

>
> From: "mumat" <csarami at gmail.com>
To: mathgroup at smc.vnet.net
> Date: 2005/09/16 Fri AM 03:50:52 EDT
> Subject: [mg60466] [mg60450] Complete solution to a modular System of equations
>
> 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?
>