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?
>
>
> regards,
>
>
>

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

```

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