  
  [1X1 [33X[0;0YDefinitions[133X[101X
  
  
  [1X1.1 [33X[0;0YDifference Sets[133X[101X
  
  [33X[0;0YA [22X⟨ v, k, λ ⟩[122X-difference set is a nonempty proper subset [22XD[122X of a finite group
  [22XG[122X  such  that  [22X|G|  =  v[122X,  [22X|D| = k[122X, and each nonidentity element of [22XG[122X can be
  written  as  [22Xd_id_j^-1[122X  for  [22Xd_i,  d_j  ∈ D[122X in exactly [22Xλ[122X different ways. The
  standard  example  is  the  [22X⟨ 7, 3, 1⟩[122X-difference set [22X{1, 2, 4}[122X of the group
  [22XZ/7Z[122X  under  addition.  Additionally,  it can easily be shown that every one
  element  subset  of  a  group is a difference set, and the complement of any
  difference set is also a difference set.[133X
  
  [33X[0;0YWe will often abuse notation and let [22XD[122X denote both the set [22XD[122X and the element[133X
  
  
  [24X[33X[0;6YD = \sum_{d \in D} d[133X
  
  [124X
  
  [33X[0;0Yof the group ring [22XZ[G][122X. Then define[133X
  
  
  [24X[33X[0;6YgD = \sum_{d \in D} gd,[133X
  
  [124X
  
  
  [24X[33X[0;6YD^\phi = \sum_{d \in D} \phi(d),[133X
  
  [124X
  
  
  [24X[33X[0;6YD^{(-1)} = \sum_{d \in D} d^{-1},[133X
  
  [124X
  
  [33X[0;0Ywhere  [22Xg  ∈  G[122X and [22Xϕ[122X is a homomorphism with domain [22XG[122X. Using this notation, a
  difference  set  in [22XG[122X is an element of the group ring [22XZ[G][122X with coefficients
  from  [22X{0,  1}[122X  such  that  [22XDD^(-1)  =  (k-λ)  + λ G[122X, where by convention the
  isolated coefficients [22X(k-λ)[122X are assumed to be coefficients of the identity.[133X
  
  [33X[0;0YTwo  difference sets [22XD_1, D_2[122X are equivalent if both are in the same group [22XG[122X
  and [22XD_1 = gD_2^ϕ[122X for some [22Xg ∈ G[122X and [22Xϕ ∈ mathrmAut(G)[122X. In other words, [22XD_1[122X is
  equivalent  to  [22XD_2[122X  if  [22XD_1[122X  can  be  mapped  to  [22XD_2[122X  by  translation  and
  automorphism  in the group [22XG[122X. We say [22XD_1, D_2[122X are translationally equivalent
  if  they are equivalent solely by translation, meaning [22XD_1 = gD_2[122X for some [22Xg
  ∈ G[122X.[133X
  
  [33X[0;0YIn  the  package,  difference  sets  are  stored  as  lists of integers that
  represent  the  index  of the elements in the difference set as found in the
  list  of all elements in the group returned by the [5XGAP[105X function [10XElements(G)[110X.
  For  example,  the  difference set [10X[1, 3, 6, 9, 11, 13][110X in [10XSmallGroup(16, 5)[110X
  really  consists of the first, third, sixth, ninth, eleventh, and thirteenth
  elements  of the list returned by [10XElements(SmallGroup(16, 5))[110X. When given as
  arguments,  difference  sets  in the package are never assumed to be sorted,
  but many functions will return difference sets in sorted order since sorting
  is used internally.[133X
  
  
  [1X1.2 [33X[0;0YDifference Sums[133X[101X
  
  [33X[0;0YA [22X⟨ v, k, λ ⟩[122X-difference sum in a group [22XG[122X modulo its normal subgroup [22XN[122X is an
  element  [22XS[122X  of  the group ring [22XZ[G/N][122X such that [22XSS^(-1) = (k - λ) + λ |N|G/N[122X
  and  the  coefficients  of [22XS[122X have values in [22X{0, 1, dots, |N|}[122X. Note that the
  original  [22XG[122X  and  [22XN[122X  are included in the definition, so it makes no sense to
  talk  about  a  difference  sum  in  some  arbitrary  group [22XH[122X. The size of a
  difference  sum  is  the  sum  of  its  coefficients,  and  by  defining the
  complement  of  [22XS[122X  to  be  [22X|N|G/N - S[122X we can see that, similar to difference
  sets, size one sums and complements of difference sums are always difference
  sums.[133X
  
  [33X[0;0YTwo  difference sums [22XS_1, S_2[122X are equivalent if both are in the same group [22XG[122X
  mod  its  normal  subgroup  [22XN[122X  and  [22XS_1  =  gS_2^ϕ[122X for some [22Xg ∈ G/N[122X and [22Xϕ[122X an
  automorphism  of  [22XG/N[122X  induced  by  an  automorphism of [22XG[122X. Note that not all
  automorphisms  of  [22XG/N[122X  are induced by automorphisms of [22XG[122X, so our definition
  here is more restrictive than perhaps expected. As with difference sets, the
  sums [22XS_1, S_2[122X are translationally equivalent if [22XS_1 = gS_2[122X for some [22Xg ∈ G/N[122X.[133X
  
  [33X[0;0YIn  the  package,  difference  sums  are  stored  as  lists of integers that
  represent  the  values  of the coefficients of the group ring elements, with
  position  in  the  list  given  by  the position of the coset in the list of
  elements  returned  by  the  [5XGAP[105X  function  [10XElements(G/N)[110X.  For example, the
  difference  sum  [10X[2,  4][110X  in  [10XG := SmallGroup(16, 5)[110X mod its normal subgroup
  [10XSubgroup(G,  [G.2,  G.3,  G.4])[110X has coefficient 2 on the identity coset, and
  coefficient 4 on the nonidentity coset.[133X
  
  [33X[0;0YDifference  sums  can  be thought of as a generalization of difference sets.
  More  importantly,  however,  difference sums can be thought of as images of
  difference  sets in quotients of the original group. In particular, if [22Xθ : G
  ->  G/N[122X  is the natural projection, then for any difference set [22XD[122X in [22XZ[G][122X we
  have  a  difference sum [22XD^θ[122X in [22XG[122X modulo its normal subgroup [22XN[122X. Additionally,
  difference  sums  induce  other difference sums in any further quotient. The
  fundamental  idea  of  the  algorithm in this package is that we can reverse
  this process. Starting with [22XG[122X mod [22XG[122X, where the only difference sum of size [22Xk[122X
  is  [10X[k][110X,  we  can  successively  refine  this  difference sum up a series of
  quotients  of  [22XG[122X  until  reaching  [22XG[122X  itself.  In each step we enumerate all
  preimages  of  the  difference  sums  and  remove  preimages  that  are  not
  difference  sums themselves. In the final step we refine to difference sets.
  Furthermore,   since   equivalent   difference  sums  will  have  equivalent
  collections  of difference sets as preimages, in each step we remove all but
  one  representative  of  each  equivalence  class  from our collection. This
  method dramatically decreases the search space for an exhaustive enumeration
  of all difference sets up to equivalence in [22XG[122X.[133X
  
