score:6

Accepted answer 
return Enum
    .GetValues( typeof(AnimalCodeType) )
    .Cast<AnimalCodeType>()
    .Where( v => (int)v > 0 )
    .Select( v => v.ToString() )
    .ToList();

score:7

This gives you all but the ant (recommended way):

var animalsWithoutAnt = Enum.GetValues(typeof(AnimalCodeType)).Cast<AnimalCodeType>()
    .Where(act => act != AnimalCodeType.Ant)
    .ToList();

or by using the int-value:

var animalsWithoutAnt = Enum.GetValues(typeof(AnimalCodeType)).Cast<AnimalCodeType>()
   .Where(act => (int)act != 0)
   .ToList();

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