Combinatorial generation via permutation languages

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an $n$-element set by adjacent transpositions; the binary reflected Gray code to generate all $n$-bit strings by flipping a single bit in each step; the Gray code for generating all $n$-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an $n$-element ground set by element exchanges due to Kaye. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We thus also obtain new Gray code algorithms for the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into $n$ rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group~$S_n$. Recently, Pilaud and Santos realized all those lattice congruences as $(n-1)$-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope

Generator Matrix Based Search for Extremal Self-Dual Binary Error-Correcting Codes

Self-dual doubly even linear binary error-correcting codes, often referred to as Type II codes, are codes closely related to many combinatorial structures such as 5-designs. Extremal codes are codes that have the largest possible minimum distance for a given length and dimension. The existence of an extremal (72,36,16) Type II code is still open. Previous results show that the automorphism group of a putative code C with the aforementioned properties has order 5 or dividing 24. In this work, we present a method and the results of an exhaustive search showing that such a code C cannot admit an automorphism group Z6. In addition, we present so far unpublished construction of the extended Golay code by P. Becker. We generalize the notion and provide example of another Type II code that can be obtained in this fashion. Consequently, we relate Becker's construction to the construction of binary Type II codes from codes over GF(2^r) via the Gray map