Olli Pottonen

The Perfect Binary One-Error-Correcting Codes of Length

A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 is presented. There are 5983 such inequivalent perfect codes and 2165 extended perfect codes. Efficient generation of these codes relies on the recent classification of Steiner quadruple systems of order 16. Utilizing a result of Blackmore, the optimal binary one-error-correcting codes of length 14 and the (15, 1024, 4) codes are also classified; there are 38 408 and 5983 such codes, respectively.

15

The metric dimension of a graph G is the size of a smallest subset L⊆V(G) such that for any x,y∈V(G) there is a z∈L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, the computational complexity of determining the metric dimension of a graph is still very unclear. Essentially, we only know the problem to be NP-hard for general graphs, to be polynomial-time solvable on trees, and to have a logn-approximation algorithm for general graphs. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on bounded-degree planar graphs is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.

: Part I—Classification

A complete classification of the perfect binary one-error-correcting codes of length 15, as well as their extensions of length 16, was recently carried out in [P. R. J. O¿stergård and O. Pottonen, ¿The perfect binary one-error-correcting codes of length 15: Part I-Classification,¿ IEEE Trans. Inf. Theory vol. 55, pp. 4657-4660, 2009]. In the current accompanying work, the classified codes are studied in great detail, and their main properties are tabulated. The results include the fact that 33 of the 80 Steiner triple systems of order 15 occur in such codes. Further understanding is gained on full-rank codes via switching, as it turns out that all but two full-rank codes can be obtained through a series of such transformations from the Hamming code. Other topics studied include (non)systematic codes, embedded one-error-correcting codes, and defining sets of codes. A classification of certain mixed perfect codes is also obtained.

On the complexity of metric dimension

The Steiner quadruple systems of order 16 are classified up to isomoiphism by means of an exhaustive computer search. The number of isomorphism classes of such designs is 1,054,163. Properties of the designs--including the orders of the automorphism groups and the structures of the derived Steiner triple systems of order 15--are tabulated. A consistency check based on double counting is carried out to gain confidence in the correctness of the classification.

The Perfect Binary One-Error-Correcting Codes of Length 15: Part II—Properties

The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect binary one-error-correcting code from its minimum distance graph is presented. Consequently, inequivalent such codes have nonisomorphic minimum distance graphs. Moreover, it is shown that the automorphism group of a minimum distance graph is isomorphic to that of the corresponding code.

The Steiner quadruple systems of order 16

If a Steiner system S(4,5,17) exists, it would contain derived S(3,4,16) designs. By relying on a recent classification of the S(3,4,16), an exhaustive computer search for S(4,5,17) is carried out. The search shows that no S(4,5,17) exists, thereby ruling out the existence of Steiner systems S(t,t+1,t+13) for t>=4.

Reconstructing Extended Perfect Binary One-Error-Correcting Codes From Their Minimum Distance Graphs

Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length

There exists no Steiner system S(4,5,17)

2^m-4

On Optimal Binary One-Error-Correcting Codes of Lengths

and

2^{m}-4

2^m-3