Ticket #14562: trac_14562.patch

doc/en/reference/combinat/designs.rst

@@ -8 +8 @@
    ../sage/combinat/designs/covering_design
    ../sage/combinat/designs/ext_rep
    ../sage/combinat/designs/incidence_structures
+   ../sage/combinat/designs/steiner_quadruple_systems
    ../sage/combinat/designs/design_catalog
+

sage/combinat/designs/all.py

@@ -34 +34 @@
                             "typing designs.best_known_covering_design_from_LJCR"))
 
 del deprecated_callable_import
+
+import sage.combinat.designs.steiner_quadruple_systems

sage/combinat/designs/design_catalog.py

@@ -42 +42 @@
     :meth:`~sage.combinat.designs.block_design.WittDesign`
     :meth:`~sage.combinat.designs.block_design.HadamardDesign`
     :meth:`~sage.combinat.designs.block_design.steiner_triple_system`
+    :meth:`~sage.combinat.designs.block_design.steiner_quadruple_system`
 
 And the :meth:`designs.best_known_covering_design_from_LJCR
 <sage.combinat.designs.covering_design.best_known_covering_design_www>` function
@@ -62 +63 @@
                                                 HadamardDesign,
                                                 steiner_triple_system)
 
+from sage.combinat.designs.steiner_quadruple_systems import steiner_quadruple_system
+
 from sage.combinat.designs.covering_design import best_known_covering_design_www as best_known_covering_design_from_LJCR

new file sage/combinat/designs/steiner_quadruple_systems.py

@@ +1 @@
+r"""
+Steiner Quadruple Systems
+
+A Steiner Quadruple System on `n` points is a family `SQS_n \subset \binom {[n]}
+4` of `4`-sets, such that any set `S\subset [n]` of size three is a subset of
+exactly one member of `SQS_n`.
+
+This module implements Haim Hanani's constructive proof that a Steiner Quadruple
+System exists if and only if `n\equiv 2,4[12]`. Hanani's proof consists in 6
+different constructions that build a large Steiner Quadruple System from a smaller
+one, and though it does not give a very clear understanding of why it works (to say the
+least)... it does !
+
+The constructions have been implemented while reading two papers simultaneously,
+for one of them sometimes provides the informations that the other one does
+not. The first one is Haim Hanani's original paper [Hanani60]_, and the other
+one is a paper from Horan and Hurlbert which goes through all constructions
+[HH12]_.
+
+It can be used through the ``designs`` object::
+
+    sage: designs.steiner_quadruple_system(8)
+    ((0, 1, 2, 3), (0, 1, 6, 7), (0, 5, 2, 7), (0, 5, 6, 3), (4, 1, 2, 7),
+    (4, 1, 6, 3), (4, 5, 2, 3), (4, 5, 6, 7), (0, 1, 4, 5), (0, 2, 4, 6),
+    (0, 3, 4, 7), (1, 2, 5, 6), (1, 3, 5, 7), (2, 3, 6, 7))
+
+REFERENCES:
+
+.. [Hanani60] Haim Hanani,
+  On quadruple systems,
+  pages 145--157, vol. 12,
+  Canadadian Journal of Mathematics,
+  1960
+  http://cms.math.ca/cjm/v12/cjm1960v12.0145-0157.pdf
+
+.. [HH12] Victoria Horan and Glenn Hurlbert,
+  Overlap Cycles for Steiner Quadruple Systems,
+  2012,
+  http://arxiv.org/abs/1204.3215
+
+AUTHORS:
+
+- Nathann Cohen (May 2013, while listening to "*Le Blues Du Pauvre Delahaye*")
+
+Index
+-----
+
+This module's main function is the following :
+
+.. csv-table::
+    :class: contentstable
+    :widths: 15, 20, 65
+    :delim: |
+
+    | :func:`steiner_quadruple_system` | Returns a Steiner Quadruple System on `n` points
+
+This function redistributes its work among 6 constructions :
+
+.. csv-table::
+    :class: contentstable
+    :widths: 15, 20, 65
+    :delim: |
+
+    Construction `1` | :func:`two_n`                | Returns a Steiner Quadruple System on `2n` points
+    Construction `2` | :func:`three_n_minus_two`    | Returns a Steiner Quadruple System on `3n-2` points
+    Construction `3` | :func:`three_n_minus_height` | Returns a Steiner Quadruple System on `3n-8` points
+    Construction `4` | :func:`three_n_minus_four`   | Returns a Steiner Quadruple System on `3n-4` points
+    Construction `5` | :func:`four_n_minus_six`     | Returns a Steiner Quadruple System on `4n-6` points
+    Construction `6` | :func:`twelve_n_minus_ten`   | Returns a Steiner Quadruple System on `12n-10` points
+
+It also defines two specific Steiner Quadruple Systems that the constructions
+require, i.e.`SQS_{14}` and `SQS_{38}` as well as the systems of pairs
+`P_{\alpha}(m)` and `\overline P_{\alpha}(m)` (see [Hanani60]_).
+
+Functions
+---------
+"""
+from sage.misc.cachefunc import cached_function
+
+# Construction 1
+def two_n(n,B):
+    r"""
+    Returns a Steiner Quadruple System on `2n` points.
+
+    INPUT:
+
+    - ``n`` (integer)
+
+    - ``B`` -- A Steiner Quadruple System on `n` points.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import two_n, is_steiner_quadruple_system
+        sage: for n in xrange(4, 30):
+        ....:     if (n%6) in [2,4]:
+        ....:         sqs = designs.steiner_quadruple_system(n)
+        ....:         if not is_steiner_quadruple_system(2*n, two_n(n, sqs)):
+        ....:             print "Something is wrong !"
+
+    """
+    Y = []
+
+    # Line 1
+    for x,y,z,t in B:
+        for a in xrange(2):
+            for b in xrange(2):
+                for c in xrange(2):
+                    d = (a+b+c)%2
+                    Y.append((x+a*n,y+b*n,z+c*n,t+d*n))
+
+    # Line 2
+    for j in xrange(n):
+        for jj in xrange(j+1,n):
+            Y.append((j,jj,n+j,n+jj))
+
+    return tuple(Y)
+
+# Construction 2
+def three_n_minus_two(n,B):
+    """
+    Returns a Steiner Quadruple System on `3n-2` points.
+
+    INPUT:
+
+    - ``n`` (integer)
+
+    - ``B`` -- A Steiner Quadruple System on `n` points.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import three_n_minus_two, is_steiner_quadruple_system
+        sage: for n in xrange(4, 30):
+        ....:     if (n%6) in [2,4]:
+        ....:         sqs = designs.steiner_quadruple_system(n)
+        ....:         if not is_steiner_quadruple_system(3*n-2, three_n_minus_two(n, sqs)):
+        ....:             print "Something is wrong !"
+
+    """
+    A = n-1
+    Y = []
+    # relabel function
+    r = lambda i,x : (i%3)*(n-1)+x
+    for x,y,z,t in B:
+        if t == A:
+            # Line 2.
+            for a in xrange(3):
+                for b in xrange(3):
+                    c = -(a+b)%3
+                    Y.append((r(a,x),r(b,y),r(c,z),3*n-3))
+
+            # Line 3.
+            Y.extend([(r(i,x),r(i,y),r(i+1,z),r(i+2,z)) for i in xrange(3)])
+            Y.extend([(r(i,x),r(i,z),r(i+1,y),r(i+2,y)) for i in xrange(3)])
+            Y.extend([(r(i,y),r(i,z),r(i+1,x),r(i+2,x)) for i in xrange(3)])
+
+        else:
+            # Line 1.
+            for a in xrange(3):
+                for b in xrange(3):
+                    for c in xrange(3):
+                      d = -(a+b+c)%3
+                      Y.append((r(a,x),r(b,y),r(c,z),r(d,t)))
+
+    # Line 4.
+    for j in xrange(n-1):
+        for jj in xrange(j+1,n-1):
+            Y.extend([(r(i,j),r(i,jj),r(i+1,j),r(i+1,jj)) for i in xrange(3)])
+
+    # Line 5.
+    for j in xrange(n-1):
+        Y.append((r(0,j),r(1,j),r(2,j),3*n-3))
+
+    Y = tuple(map(tuple,map(sorted,Y)))
+    return Y
+
+# Construction 3
+def three_n_minus_height(n, B):
+    """
+    Returns a Steiner Quadruple System on `3n-8` points.
+
+    INPUT:
+
+    - ``n`` -- an integer such that `n\equiv 2[12]`.
+
+    - ``B`` -- A Steiner Quadruple System on `n` points.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import three_n_minus_height, is_steiner_quadruple_system
+        sage: for n in xrange(4, 30):
+        ....:     if (n%12) == 2:
+        ....:         sqs = designs.steiner_quadruple_system(n)
+        ....:         if not is_steiner_quadruple_system(3*n-8, three_n_minus_height(n, sqs)):
+        ....:             print "Something is wrong !"
+
+    """
+
+    if (n%12) != 2:
+        raise ValueError("n must be equal to 2 mod 12")
+
+    B = relabel_system(n, B)
+    r = lambda i,x : (i%3)*(n-4)+(x%(n-4))
+
+    # Line 1.
+    Y = [[x+2*(n-4) for x in B[0]]]
+
+    # Line 2.
+    for s in B[1:]:
+        for i in xrange(3):
+            Y.append([r(i,x) if x<= n-5 else x+2*(n-4) for x in s])
+
+
+    # Line 3.
+    for a in xrange(4):
+        for aa in xrange(n-4):
+            for aaa in xrange(n-4):
+                aaaa = -(a+aa+aaa)%(n-4)
+                Y.append((r(0,aa),r(1,aaa), r(2,aaaa),3*(n-4)+a))
+
+
+    # Line 4.
+    k = (n-14)/12
+    for i in xrange(3):
+        for b in xrange(n-4):
+            for bb in xrange(n-4):
+                bbb = -(b+bb)%(n-4)
+                for d in xrange(2*k+1):
+                    Y.append((r(i+2,bbb), r(i, b+2*k+1+i*(4*k+2)-d) , r(i, b+2*k+2+i*(4*k+2)+d), r(i+1,bb)))
+
+
+
+    # Line 5.
+    for i in xrange(3):
+        for alpha in xrange(4*k+2, 12*k+9):
+            for ra,sa in P(alpha,6*k+5):
+                for raa,saa in P(alpha,6*k+5):
+                    Y.append(tuple(sorted((r(i,ra),r(i,sa),r(i+1,raa), r(i+1,saa)))))
+
+
+    Y = tuple(map(tuple,map(sorted,Y)))
+    return Y
+
+# Construction 4
+def three_n_minus_four(n, B):
+    """
+    Returns a Steiner Quadruple System on `3n-4` points.
+
+    INPUT:
+
+    - ``n`` -- an integer such that `n\equiv 10[12]`
+
+    - ``B`` -- A Steiner Quadruple System on `n` points.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import three_n_minus_four, is_steiner_quadruple_system
+        sage: for n in xrange(4, 30):
+        ....:     if n%12 == 10:
+        ....:         sqs = designs.steiner_quadruple_system(n)
+        ....:         if not is_steiner_quadruple_system(3*n-4, three_n_minus_four(n, sqs)):
+        ....:             print "Something is wrong !"
+
+    """
+    if n%12 != 10:
+        raise ValueError("n must be equal to 10 mod 12")
+
+    B = relabel_system(n, B)
+    r = lambda i,x : (i%3)*(n-2)+(x%(n-2))
+
+    # Line 1/2.
+    Y = []
+    for s in B:
+        for i in xrange(3):
+            Y.append(tuple(r(i,x) if x<= n-3 else x+2*(n-2) for x in s ))
+
+    # Line 3.
+    for a in xrange(2):
+        for aa in xrange(n-2):
+            for aaa in xrange(n-2):
+                aaaa= -(a+aa+aaa)%(n-2)
+                Y.append((r(0,aa),r(1,aaa), r(2,aaaa),3*(n-2)+a))
+
+    # Line 4.
+    k = (n-10)/12
+    for i in xrange(3):
+        for b in xrange(n-2):
+            for bb in xrange(n-2):
+                bbb = -(b+bb)%(n-2)
+                for d in xrange(2*k+1):
+                    Y.append((r(i+2,bbb), r(i, b+2*k+1+i*(4*k+2)-d) , r(i, b+2*k+2+i*(4*k+2)+d), r(i+1,bb)))
+
+    # Line 5.
+    from sage.graphs.graph_coloring import round_robin
+    one_factorization = round_robin(2*(6*k+4)).edges()
+    color_classes = [[] for j in xrange(2*(6*k+4)-1)]
+    for u,v,l in one_factorization:
+        color_classes[l].append((u,v))
+
+    for i in xrange(3):
+        for alpha in xrange(4*k+2, 12*k+6+1):
+            for ra,sa in P(alpha, 6*k+4):
+                for raa,saa in P(alpha, 6*k+4):
+                    Y.append(tuple(sorted((r(i,ra),r(i,sa),r(i+1,raa), r(i+1,saa)))))
+
+    Y = tuple(map(tuple,map(sorted,Y)))
+    return Y
+
+# Construction 5
+def four_n_minus_six(n, B):
+    """
+    Returns a Steiner Quadruple System on `4n-6` points.
+
+    INPUT:
+
+    - ``n`` (integer)
+
+    - ``B`` -- A Steiner Quadruple System on `n` points.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import four_n_minus_six, is_steiner_quadruple_system
+        sage: for n in xrange(4, 20):
+        ....:     if (n%6) in [2,4]:
+        ....:         sqs = designs.steiner_quadruple_system(n)
+        ....:         if not is_steiner_quadruple_system(4*n-6, four_n_minus_six(n, sqs)):
+        ....:             print "Something is wrong !"
+
+    """
+
+    f = n-2
+    r = lambda i,ii,x : (2*(i%2)+(ii%2))*(n-2)+(x)%(n-2)
+
+    # Line 1.
+    Y = []
+    for s in B:
+        for i in xrange(2):
+            for ii in xrange(2):
+                Y.append(tuple(r(i,ii,x) if x<= n-3 else x+3*(n-2) for x in s ))
+
+    # Line 2/3/4/5
+    k = f/2
+    for l in xrange(2):
+        for eps in xrange(2):
+            for c in xrange(k):
+                for cc in xrange(k):
+                    ccc = -(c+cc)%k
+                    Y.append((4*(n-2)+l,r(0,0,2*c),r(0,1,2*cc-eps),r(1,eps,2*ccc+l)))
+                    Y.append((4*(n-2)+l,r(0,0,2*c+1),r(0,1,2*cc-1-eps),r(1,eps,2*ccc+1-l)))
+                    Y.append((4*(n-2)+l,r(1,0,2*c),r(1,1,2*cc-eps),r(0,eps,2*ccc+1-l)))
+                    Y.append((4*(n-2)+l,r(1,0,2*c+1),r(1,1,2*cc-1-eps),r(0,eps,2*ccc+l)))
+
+    # Line 6/7
+    for h in xrange(2):
+        for eps in xrange(2):
+            for ccc in xrange(k):
+                assert len(barP(ccc,k)) == k-1
+                for rc,sc in barP(ccc,k):
+                    for c in xrange(k):
+                        cc = -(c+ccc)%k
+                        Y.append((r(h,0,2*c+eps),r(h,1,2*cc-eps),r(h+1,0,rc),r(h+1,0,sc)))
+                        Y.append((r(h,0,2*c-1+eps),r(h,1,2*cc-eps),r(h+1,1,rc),r(h+1,1,sc)))
+
+
+
+    # Line 8/9
+    for h in xrange(2):
+        for eps in xrange(2):
+            for ccc in xrange(k):
+                for rc,sc in barP(k+ccc,k):
+                    for c in xrange(k):
+                        cc = -(c+ccc)%k
+                        Y.append((r(h,0,2*c+eps),r(h,1,2*cc-eps),r(h+1,1,rc),r(h+1,1,sc)))
+                        Y.append((r(h,0,2*c-1+eps),r(h,1,2*cc-eps),r(h+1,0,rc),r(h+1,0,sc)))
+
+
+    # Line 10
+    for h in xrange(2):
+        for alpha in xrange(n-3):
+            for ra,sa in P(alpha,k):
+                for raa,saa in P(alpha,k):
+                    Y.append((r(h,0,ra),r(h,0,sa),r(h,1,raa),r(h,1,saa)))
+
+    Y = tuple(map(tuple,map(sorted,Y)))
+    return Y
+
+# Construction 6
+def twelve_n_minus_ten(n, B):
+    """
+    Returns a Steiner Quadruple System on `12n-6` points.
+
+    INPUT:
+
+    - ``n`` (integer)
+
+    - ``B`` -- A Steiner Quadruple System on `n` points.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import twelve_n_minus_ten, is_steiner_quadruple_system
+        sage: for n in xrange(4, 15):
+        ....:     if (n%6) in [2,4]:
+        ....:         sqs = designs.steiner_quadruple_system(n)
+        ....:         if not is_steiner_quadruple_system(12*n-10, twelve_n_minus_ten(n, sqs)):
+        ....:             print "Something is wrong !"
+
+    """
+
+    B14 = steiner_quadruple_system(14)
+    r = lambda i,x : i%(n-1)+(x%12)*(n-1)
+    k = n/2
+
+    # Line 1.
+    Y = []
+    for s in B14:
+        for i in xrange(n-1):
+            Y.append(tuple(r(i,x) if x<= 11 else r(n-2,11)+x-11 for x in s ))
+
+    for s in B:
+        if s[-1] == n-1:
+            u,v,w,B = s
+            dd = {0:u,1:v,2:w}
+            d = lambda x:dd[x%3]
+            for b in xrange(12):
+                for bb in xrange(12):
+                    bbb = -(b+bb)%12
+                    for h in xrange(2):
+                        # Line 2
+                        Y.append((r(n-2,11)+1+h,r(u,b),r(v,bb),r(w,bbb+3*h)))
+
+                    for i in xrange(3):
+                        # Line 38.3
+                        Y.append((  r(d(i),b+4+i), r(d(i),b+7+i), r(d(i+1),bb), r(d(i+2),bbb)))
+
+            for j in xrange(12):
+                for eps in xrange(2):
+                    for i in xrange(3):
+                        # Line 38.4-38.7
+                        Y.append((  r(d(i),j), r(d(i+1),j+6*eps  ), r(d(i+2),6*eps-2*j+1), r(d(i+2),6*eps-2*j-1)))
+                        Y.append((  r(d(i),j), r(d(i+1),j+6*eps  ), r(d(i+2),6*eps-2*j+2), r(d(i+2),6*eps-2*j-2)))
+                        Y.append((  r(d(i),j), r(d(i+1),j+6*eps-3), r(d(i+2),6*eps-2*j+1), r(d(i+2),6*eps-2*j+2)))
+                        Y.append((  r(d(i),j), r(d(i+1),j+6*eps+3), r(d(i+2),6*eps-2*j-1), r(d(i+2),6*eps-2*j-2)))
+
+            for j in xrange(6):
+                for i in xrange(3):
+                    for eps in xrange(2):
+                        # Line 38.8
+                        Y.append((  r(d(i),j), r(d(i),j+6), r(d(i+1),j+3*eps), r(d(i+1),j+6+3*eps)))
+
+            for j in xrange(12):
+                for i in xrange(3):
+                    for eps in xrange(4):
+                        # Line 38.11
+                        Y.append((  r(d(i),j), r(d(i),j+1), r(d(i+1),j+3*eps), r(d(i+1),j+3*eps+1)))
+                        # Line 38.12
+                        Y.append((  r(d(i),j), r(d(i),j+2), r(d(i+1),j+3*eps), r(d(i+1),j+3*eps+2)))
+                        # Line 38.13
+                        Y.append((  r(d(i),j), r(d(i),j+4), r(d(i+1),j+3*eps), r(d(i+1),j+3*eps+4)))
+
+            for alpha in [4,5]:
+                for ra,sa in P(alpha,6):
+                    for raa,saa in P(alpha,6):
+                        for i in xrange(3):
+                            for ii in xrange(i+1,3):
+                                # Line 38.14
+                                Y.append((  r(d(i),ra), r(d(i),sa), r(d(ii),raa), r(d(ii),saa)))
+
+            for g in xrange(6):
+                for eps in xrange(2):
+                    for i in xrange(3):
+                        for ii in xrange(3):
+                            if i == ii:
+                                continue
+                            # Line 38.9
+                            Y.append((  r(d(i),2*g+3*eps), r(d(i),2*g+6+3*eps), r(d(ii),2*g+1), r(d(ii),2*g+5)))
+                            # Line 38.10
+                            Y.append((  r(d(i),2*g+3*eps), r(d(i),2*g+6+3*eps), r(d(ii),2*g+2), r(d(ii),2*g+4)))
+
+        else:
+            x,y,z,t = s
+            for a in xrange(12):
+                for aa in xrange(12):
+                    for aaa in xrange(12):
+                        aaaa = -(a+aa+aaa)%12
+                        # Line 3
+                        Y.append((r(x,a), r(y,aa), r(z,aaa), r(t,aaaa)))
+    return Y
+
+def relabel_system(n,B):
+    r"""
+    Relabels the set so that `\{n-4, n-3, n-2, n-1\}` is in `B`.
+
+    INPUT:
+
+    - ``n`` -- an integer
+
+    - ``B`` -- a list of 4-uples on `0,...,n-1`.
+
+    EXAMPLE::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import relabel_system
+        sage: designs.steiner_quadruple_system(8)
+        ((0, 1, 2, 3), (0, 1, 6, 7), (0, 5, 2, 7), (0, 5, 6, 3), (4, 1, 2, 7),
+        (4, 1, 6, 3), (4, 5, 2, 3), (4, 5, 6, 7), (0, 1, 4, 5), (0, 2, 4, 6),
+        (0, 3, 4, 7), (1, 2, 5, 6), (1, 3, 5, 7), (2, 3, 6, 7))
+        sage: relabel_system(8,designs.steiner_quadruple_system(8))
+        ((4, 5, 6, 7), (0, 1, 4, 5), (1, 2, 4, 6), (0, 2, 4, 7), (1, 3, 5, 6),
+        (0, 3, 5, 7), (2, 3, 6, 7), (0, 1, 2, 3), (2, 3, 4, 5), (0, 3, 4, 6),
+        (1, 3, 4, 7), (0, 2, 5, 6), (1, 2, 5, 7), (0, 1, 6, 7))
+    """
+
+    label = {
+        B[0][0] : n-4,
+        B[0][1] : n-3,
+        B[0][2] : n-2,
+        B[0][3] : n-1
+        }
+
+    def get_label(x):
+        if x in label:
+            return label[x]
+        else:
+            total = len(label)-4
+            label[x] = total
+            return total
+
+    B = tuple([tuple(sorted(map(get_label,s))) for s in B])
+    return B
+
+def P(alpha, m):
+    r"""
+    Returns the collection of pairs `P_{\alpha}(m)`
+
+    For more information on this system, see [Hanani60]_.
+
+    EXAMPLE::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import P
+        sage: P(3,4)
+        [(0, 5), (2, 7), (4, 1), (6, 3)]
+    """
+    if alpha >= 2*m-1:
+        raise Exception
+    if m%2==0:
+        if alpha < m:
+            if alpha%2 == 0:
+                b = alpha/2
+                return [(2*a, (2*a + 2*b + 1)%(2*m)) for a in xrange(m)]
+            else:
+                b = (alpha-1)/2
+                return [(2*a, (2*a - 2*b - 1)%(2*m)) for a in xrange(m)]
+        else:
+            y = alpha - m
+            pairs  = [(b,(2*y-b)%(2*m)) for b in xrange(y)]
+            pairs += [(c,(2*m+2*y-c-2)%(2*m)) for c in xrange(2*y+1,m+y-1)]
+            pairs += [(2*m+int(-1.5-.5*(-1)**y),y),(2*m+int(-1.5+.5*(-1)**y),m+y-1)]
+            return pairs
+    else:
+        if alpha < m-1:
+            if alpha % 2 == 0:
+                b = alpha/2
+                return [(2*a,(2*a+2*b+1)%(2*m)) for a in xrange(m)]
+            else:
+                b = (alpha-1)/2
+                return [(2*a,(2*a-2*b-1)%(2*m)) for a in xrange(m)]
+        else:
+            y = alpha-m+1
+            pairs  = [(b,2*y-b) for b in xrange(y)]
+            pairs += [(c,2*m+2*y-c) for c in xrange(2*y+1,m+y)]
+            pairs += [(y,m+y)]
+            return pairs
+
+def _missing_pair(n,l):
+    r"""
+    Returns the smallest `(x,x+1)` that is not contained in `l`.
+
+    EXAMPLE::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import _missing_pair
+        sage: _missing_pair(6, [(0,1), (4,5)])
+        (2, 3)
+    """
+    l = [x for X in l for x in X]
+    for x in xrange(n):
+        if not x in l:
+            break
+
+    assert not x in l
+    assert not x+1 in l
+    return (x,x+1)
+
+def barP(eps, m):
+    r"""
+    Returns the collection of pairs `\overline P_{\alpha}(m)`
+
+    For more information on this system, see [Hanani60]_.
+
+    EXAMPLE::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import barP
+        sage: barP(3,4)
+        [(0, 4), (3, 5), (1, 2)]
+    """
+    return barP_system(m)[eps]
+
+@cached_function
+def barP_system(m):
+    r"""
+    Returns the 1-factorization of `K_{2m}` `\overline P(m)`
+
+    For more information on this system, see [Hanani60]_.
+
+    EXAMPLE::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import barP_system
+        sage: barP_system(3)
+        [[(4, 3), (2, 5)],
+        [(0, 5), (4, 1)],
+        [(0, 2), (1, 3)],
+        [(1, 5), (4, 2), (0, 3)],
+        [(0, 4), (3, 5), (1, 2)],
+        [(0, 1), (2, 3), (4, 5)]]
+    """
+    isequal = lambda e1,e2 : e1 == e2 or e1 == tuple(reversed(e2))
+    pairs = []
+    last = []
+
+    if m % 2 == 0:
+        # The first (shorter) collections of  pairs, obtained from P by removing
+        # pairs. Those are added to 'last', a new list of pairs
+        last = []
+        for n in xrange(1,(m-2)/2+1):
+            pairs.append([p for p in P(2*n,m) if not isequal(p,(2*n,(4*n+1)%(2*m)))])
+            last.append((2*n,(4*n+1)%(2*m)))
+            pairs.append([p for p in P(2*n-1,m) if not isequal(p,(2*m-2-2*n,2*m-1-4*n))])
+            last.append((2*m-2-2*n,2*m-1-4*n))
+
+        pairs.append([p for p in P(m,m) if not isequal(p,(2*m-2,0))])
+        last.append((2*m-2,0))
+        pairs.append([p for p in P(m+1,m) if not isequal(p,(2*m-1,1))])
+        last.append((2*m-1,1))
+
+        assert all(len(pp) == m-1 for pp in pairs)
+        assert len(last) == m
+
+        # Pairs of normal length
+
+        pairs.append(P(0,m))
+        pairs.append(P(m-1,m))
+
+        for alpha in xrange(m+2,2*m-1):
+            pairs.append(P(alpha,m))
+        pairs.append(last)
+
+        assert len(pairs) == 2*m
+
+        # Now the points must be relabeled
+        relabel = {}
+        for n in xrange(1,(m-2)/2+1):
+            relabel[2*n] = (4*n)%(2*m)
+            relabel[4*n+1] = (4*n+1)%(2*m)
+            relabel[2*m-2-2*n] = (4*n-2)%(2*m)
+            relabel[2*m-1-4*n] = (4*n-1)%(2*m)
+
+        relabel[2*m-2] = (1)%(2*m)
+        relabel[0] = 0
+        relabel[2*m-1] = 2*m-1
+        relabel[1] = 2*m-2
+
+    else:
+        # The first (shorter) collections of  pairs, obtained from P by removing
+        # pairs. Those are added to 'last', a new list of pairs
+
+        last = []
+        for n in xrange(0,(m-3)/2+1):
+            pairs.append([p for p in P(2*n,m) if not isequal(p,(2*n,(4*n+1)%(2*m)))])
+            last.append((2*n,(4*n+1)%(2*m)))
+            pairs.append([p for p in P(2*n+1,m) if not isequal(p,(2*m-2-2*n,2*m-3-4*n))])
+            last.append((2*m-2-2*n,2*m-3-4*n))
+
+        pairs.append([p for p in P(2*m-2,m) if not isequal(p,(m-1,2*m-1))])
+        last.append((m-1,2*m-1))
+
+        assert all(len(pp) == m-1 for pp in pairs)
+        assert len(pairs) == m
+
+        # Pairs of normal length
+
+        for alpha in xrange(m-1,2*m-2):
+            pairs.append(P(alpha,m))
+        pairs.append(last)
+
+        assert len(pairs) == 2*m
+
+        # Now the points must be relabeled
+        relabel = {}
+        for n in xrange(0,(m-3)/2+1):
+            relabel[2*n] = (4*n)%(2*m)
+            relabel[4*n+1] = (4*n+1)%(2*m)
+            relabel[2*m-2-2*n] = (4*n+2)%(2*m)
+            relabel[2*m-3-4*n] = (4*n+3)%(2*m)
+        relabel[m-1] = (2*m-2)%(2*m)
+        relabel[2*m-1] = 2*m-1
+
+    assert len(relabel) == 2*m
+    assert len(pairs) == 2*m
+
+    # Relabeling the points
+
+    pairs = [[(relabel[x],relabel[y]) for x,y in pp] for pp in pairs]
+
+    # Pairs are sorted first according to their cardinality, then using the
+    # number of the smallest point that they do NOT contain.
+    pairs.sort(key=lambda x: _missing_pair(2*m+1,x))
+
+    return pairs
+
+@cached_function
+def steiner_quadruple_system(n, check = False):
+    r"""
+    Returns a Steiner Quadruple System on `n` points.
+
+    INPUT:
+
+    - ``n`` -- an integer such that `n\equiv 2,4[12]`
+
+    - ``check`` (boolean) -- whether to check that the system is a Steiner
+      Quadruple System before returning it (`False` by default)
+
+    EXAMPLES::
+
+        sage: designs.steiner_quadruple_system(4)
+        ((0, 1, 2, 3),)
+        sage: designs.steiner_quadruple_system(8)
+        ((0, 1, 2, 3), (0, 1, 6, 7), (0, 5, 2, 7), (0, 5, 6, 3), (4, 1, 2, 7),
+        (4, 1, 6, 3), (4, 5, 2, 3), (4, 5, 6, 7), (0, 1, 4, 5), (0, 2, 4, 6),
+        (0, 3, 4, 7), (1, 2, 5, 6), (1, 3, 5, 7), (2, 3, 6, 7))
+
+    TESTS::
+
+        sage: for n in xrange(4, 100):                                      # long time
+        ....:     if (n%6) in [2,4]:                                        # long time
+        ....:         sqs = designs.steiner_quadruple_system(n, check=True) # long time
+    """
+    n = int(n)
+    if not ((n%6) in [2, 4]):
+        raise ValueError("n mod 12 must be equal to 2 or 4")
+    elif n == 4:
+        return ((0,1,2,3),)
+    elif n == 14:
+        return _SQS14()
+    elif n == 38:
+        return _SQS38()
+    elif (n%12) in [4,8]:
+        nn = n/2
+        sqs = two_n(nn,steiner_quadruple_system(nn, check = False))
+    elif (n%18) in [4,10]:
+        nn = (n+2)/3
+        sqs = three_n_minus_two(nn,steiner_quadruple_system(nn, check = False))
+    elif (n%36) == 34:
+        nn = (n+8)/3
+        sqs = three_n_minus_height(nn,steiner_quadruple_system(nn, check = False))
+    elif (n%36) == 26 :
+        nn = (n+4)/3
+        sqs = three_n_minus_four(nn,steiner_quadruple_system(nn, check = False))
+    elif (n%24) in [2,10]:
+        nn = (n+6)/4
+        sqs = four_n_minus_six(nn,steiner_quadruple_system(nn, check = False))
+    elif (n%72) in [14,38]:
+        nn = (n+10)/12
+        sqs = twelve_n_minus_ten(nn, steiner_quadruple_system(nn, check = False))
+    else:
+        raise ValueError("This shouldn't happen !")
+
+    if check:
+        if not is_steiner_quadruple_system(n, sqs):
+            raise RuntimeError("Something is very very wrong.")
+
+    return sqs
+
+def is_steiner_quadruple_system(n,B):
+    r"""
+    Tests if `B` is a Steiner Quadruple System on `0,...,n-1`.
+
+    INPUT:
+
+    - ``n`` (integer)
+
+    - ``B`` -- a list of quadruples.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import is_steiner_quadruple_system
+        sage: is_steiner_quadruple_system(8,designs.steiner_quadruple_system(8))
+        True
+    """
+    from sage.rings.arith import binomial
+    # Cardinality
+    if len(B)*4 != binomial(n,3):
+        return False
+
+    # Vertex set
+    V = set([])
+    for b in B:
+        for x in b:
+            V.add(x)
+
+    if V != set(range(n)):
+        return False
+
+    # No two 4-sets intersect on 3 elements.
+    from itertools import combinations
+    s = set([])
+    for b in B:
+        for e in combinations(b,3):
+            if frozenset(e) in s:
+                return False
+            s.add(frozenset(e))
+
+    return True
+
+
+def _SQS14():
+    r"""
+    Returns a Steiner Quadruple System on 14 points.
+
+    Obtained form the La Jolla Covering Repository.
+
+    EXAMPLE::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import is_steiner_quadruple_system, _SQS14
+        sage: is_steiner_quadruple_system(14,_SQS14())
+        True
+    """
+    return ((0, 1, 2, 5), (0, 1, 3, 6), (0, 1, 4, 13), (0, 1, 7, 10), (0, 1, 8, 9),
+            (0, 1, 11, 12), (0, 2, 3, 4), (0, 2, 6, 12), (0, 2, 7, 9), (0, 2, 8, 11),
+            (0, 2, 10, 13), (0, 3, 5, 13), (0, 3, 7, 11), (0, 3, 8, 10), (0, 3, 9, 12),
+            (0, 4, 5, 9), (0, 4, 6, 11), (0, 4, 7, 8), (0, 4, 10, 12), (0, 5, 6, 8),
+            (0, 5, 7, 12), (0, 5, 10, 11), (0, 6, 7, 13), (0, 6, 9, 10), (0, 8, 12, 13),
+            (0, 9, 11, 13), (1, 2, 3, 13), (1, 2, 4, 12), (1, 2, 6, 9), (1, 2, 7, 11),
+            (1, 2, 8, 10), (1, 3, 4, 5), (1, 3, 7, 8), (1, 3, 9, 11), (1, 3, 10, 12),
+            (1, 4, 6, 10), (1, 4, 7, 9), (1, 4, 8, 11), (1, 5, 6, 11), (1, 5, 7, 13),
+            (1, 5, 8, 12), (1, 5, 9, 10), (1, 6, 7, 12), (1, 6, 8, 13), (1, 9, 12, 13),
+            (1, 10, 11, 13), (2, 3, 5, 11), (2, 3, 6, 7), (2, 3, 8, 12), (2, 3, 9, 10),
+            (2, 4, 5, 13), (2, 4, 6, 8), (2, 4, 7, 10), (2, 4, 9, 11), (2, 5, 6, 10),
+            (2, 5, 7, 8), (2, 5, 9, 12), (2, 6, 11, 13), (2, 7, 12, 13), (2, 8, 9, 13),
+            (2, 10, 11, 12), (3, 4, 6, 9), (3, 4, 7, 12), (3, 4, 8, 13), (3, 4, 10, 11),
+            (3, 5, 6, 12), (3, 5, 7, 10), (3, 5, 8, 9), (3, 6, 8, 11), (3, 6, 10, 13),
+            (3, 7, 9, 13), (3, 11, 12, 13), (4, 5, 6, 7), (4, 5, 8, 10), (4, 5, 11, 12),
+            (4, 6, 12, 13), (4, 7, 11, 13), (4, 8, 9, 12), (4, 9, 10, 13), (5, 6, 9, 13),
+            (5, 7, 9, 11), (5, 8, 11, 13), (5, 10, 12, 13), (6, 7, 8, 9), (6, 7, 10, 11),
+            (6, 8, 10, 12), (6, 9, 11, 12), (7, 8, 10, 13), (7, 8, 11, 12), (7, 9, 10, 12),
+            (8, 9, 10, 11))
+
+def _SQS38():
+    r"""
+    Returns a Steiner Quadruple System on 14 points.
+
+    Obtained form the La Jolla Covering Repository.
+
+    EXAMPLE::
+
+        sage: from sage.combinat.designs.steiner_quadruple_systems import is_steiner_quadruple_system, _SQS38
+        sage: is_steiner_quadruple_system(38,_SQS38())
+        True
+    """
+
+# From the La Jolla Covering Repository
+    return ((0, 1, 2, 14), (0, 1, 3, 34), (0, 1, 4, 31), (0, 1, 5, 27), (0, 1, 6, 17),
+            (0, 1, 7, 12), (0, 1, 8, 36), (0, 1, 9, 10), (0, 1, 11, 18), (0, 1, 13, 37),
+            (0, 1, 15, 35), (0, 1, 16, 22), (0, 1, 19, 33), (0, 1, 20, 25), (0, 1, 21, 23),
+            (0, 1, 24, 32), (0, 1, 26, 28), (0, 1, 29, 30), (0, 2, 3, 10), (0, 2, 4, 9),
+            (0, 2, 5, 28), (0, 2, 6, 15), (0, 2, 7, 36), (0, 2, 8, 23), (0, 2, 11, 22),
+            (0, 2, 12, 13), (0, 2, 16, 25), (0, 2, 17, 18), (0, 2, 19, 30), (0, 2, 20, 35),
+            (0, 2, 21, 29), (0, 2, 24, 34), (0, 2, 26, 31), (0, 2, 27, 32), (0, 2, 33, 37),
+            (0, 3, 4, 18), (0, 3, 5, 23), (0, 3, 6, 32), (0, 3, 7, 19), (0, 3, 8, 20),
+            (0, 3, 9, 17), (0, 3, 11, 25), (0, 3, 12, 24), (0, 3, 13, 27), (0, 3, 14, 31),
+            (0, 3, 15, 22), (0, 3, 16, 28), (0, 3, 21, 33), (0, 3, 26, 36), (0, 3, 29, 35),
+            (0, 3, 30, 37), (0, 4, 5, 7), (0, 4, 6, 28), (0, 4, 8, 25), (0, 4, 10, 30),
+            (0, 4, 11, 20), (0, 4, 12, 32), (0, 4, 13, 36), (0, 4, 14, 29), (0, 4, 15, 27),
+            (0, 4, 16, 35), (0, 4, 17, 22), (0, 4, 19, 23), (0, 4, 21, 34), (0, 4, 24, 33),
+            (0, 4, 26, 37), (0, 5, 6, 24), (0, 5, 8, 26), (0, 5, 9, 29), (0, 5, 10, 20),
+            (0, 5, 11, 13), (0, 5, 12, 14), (0, 5, 15, 33), (0, 5, 16, 37), (0, 5, 17, 35),
+            (0, 5, 18, 19), (0, 5, 21, 25), (0, 5, 22, 30), (0, 5, 31, 32), (0, 5, 34, 36),
+            (0, 6, 7, 30), (0, 6, 8, 33), (0, 6, 9, 12), (0, 6, 10, 18), (0, 6, 11, 37),
+            (0, 6, 13, 31), (0, 6, 14, 35), (0, 6, 16, 29), (0, 6, 19, 25), (0, 6, 20, 27),
+            (0, 6, 21, 36), (0, 6, 22, 23), (0, 6, 26, 34), (0, 7, 8, 11), (0, 7, 9, 33),
+            (0, 7, 10, 21), (0, 7, 13, 20), (0, 7, 14, 22), (0, 7, 15, 31), (0, 7, 16, 34),
+            (0, 7, 17, 29), (0, 7, 18, 24), (0, 7, 23, 26), (0, 7, 25, 32), (0, 7, 27, 28),
+            (0, 7, 35, 37), (0, 8, 9, 37), (0, 8, 10, 27), (0, 8, 12, 18), (0, 8, 13, 30),
+            (0, 8, 14, 15), (0, 8, 16, 21), (0, 8, 17, 19), (0, 8, 22, 35), (0, 8, 24, 31),
+            (0, 8, 28, 34), (0, 8, 29, 32), (0, 9, 11, 30), (0, 9, 13, 23), (0, 9, 14, 18),
+            (0, 9, 15, 25), (0, 9, 16, 26), (0, 9, 19, 28), (0, 9, 20, 36), (0, 9, 21, 35),
+            (0, 9, 22, 24), (0, 9, 27, 31), (0, 9, 32, 34), (0, 10, 11, 36),
+            (0, 10, 12, 15), (0, 10, 13, 26), (0, 10, 14, 16), (0, 10, 17, 37),
+            (0, 10, 19, 29), (0, 10, 22, 31), (0, 10, 23, 32), (0, 10, 24, 35),
+            (0, 10, 25, 34), (0, 10, 28, 33), (0, 11, 12, 16), (0, 11, 14, 24),
+            (0, 11, 15, 26), (0, 11, 17, 31), (0, 11, 19, 21), (0, 11, 23, 34),
+            (0, 11, 27, 29), (0, 11, 28, 35), (0, 11, 32, 33), (0, 12, 17, 20),
+            (0, 12, 19, 35), (0, 12, 21, 28), (0, 12, 22, 25), (0, 12, 23, 27),
+            (0, 12, 26, 29), (0, 12, 30, 33), (0, 12, 31, 34), (0, 12, 36, 37),
+            (0, 13, 14, 33), (0, 13, 15, 29), (0, 13, 16, 24), (0, 13, 17, 21),
+            (0, 13, 18, 34), (0, 13, 19, 32), (0, 13, 22, 28), (0, 13, 25, 35),
+            (0, 14, 17, 26), (0, 14, 19, 20), (0, 14, 21, 32), (0, 14, 23, 36),
+            (0, 14, 25, 28), (0, 14, 27, 30), (0, 14, 34, 37), (0, 15, 16, 36),
+            (0, 15, 17, 23), (0, 15, 18, 20), (0, 15, 19, 34), (0, 15, 21, 37),
+            (0, 15, 24, 28), (0, 15, 30, 32), (0, 16, 17, 32), (0, 16, 18, 27),
+            (0, 16, 19, 31), (0, 16, 20, 33), (0, 16, 23, 30), (0, 17, 24, 27),
+            (0, 17, 25, 33), (0, 17, 28, 36), (0, 17, 30, 34), (0, 18, 21, 26),
+            (0, 18, 22, 29), (0, 18, 23, 28), (0, 18, 25, 31), (0, 18, 30, 35),
+            (0, 18, 32, 37), (0, 18, 33, 36), (0, 19, 22, 26), (0, 19, 24, 37),
+            (0, 19, 27, 36), (0, 20, 21, 31), (0, 20, 22, 37), (0, 20, 23, 24),
+            (0, 20, 26, 30), (0, 20, 28, 32), (0, 20, 29, 34), (0, 21, 22, 27),
+            (0, 21, 24, 30), (0, 22, 32, 36), (0, 22, 33, 34), (0, 23, 25, 29),
+            (0, 23, 31, 37), (0, 23, 33, 35), (0, 24, 25, 26), (0, 24, 29, 36),
+            (0, 25, 27, 37), (0, 25, 30, 36), (0, 26, 27, 33), (0, 26, 32, 35),
+            (0, 27, 34, 35), (0, 28, 29, 37), (0, 28, 30, 31), (0, 29, 31, 33),
+            (0, 31, 35, 36), (1, 2, 3, 15), (1, 2, 4, 35), (1, 2, 5, 32), (1, 2, 6, 28),
+            (1, 2, 7, 18), (1, 2, 8, 13), (1, 2, 9, 37), (1, 2, 10, 11), (1, 2, 12, 19),
+            (1, 2, 16, 36), (1, 2, 17, 23), (1, 2, 20, 34), (1, 2, 21, 26), (1, 2, 22, 24),
+            (1, 2, 25, 33), (1, 2, 27, 29), (1, 2, 30, 31), (1, 3, 4, 11), (1, 3, 5, 10),
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