Large sets of Kirkman triple systems of prime power sizes

Abstract

Research on the existence of large sets of Kirkman triple systems (LKTS) extends from the mid-eighteen hundreds to the present. Enlightened by known direct constructions of LKTSs, we bring forth new approaches and finally establish the existence of LKTSs of all admissible prime power sizes less than 400 only with two possible exceptions. In the process, we also employ known construction methods and draw support from efficient algorithms.

Appendices

Appendix 1

We list all the base blocks of $(q-1,3,\{2,3\},1)$ difference families over $\mathbb {Z}_{q-1}$ in Lemma 3.1 meeting all the conditions of Theorem 2.2. We also give a primitive element $\xi$ in use or its primitive polynomial f(x). Note that in each base block set the two pairs $\{0,\alpha \}$, $\{\beta ,\beta +{q-1\over 2}\}$ are listed in order; they give rise to three blocks $\{\infty _1,1,\xi ^{\alpha }\}$, $\{\infty _2,-1,-\xi ^{\alpha }\}$, and $\{0,\xi ^\beta ,-\xi ^\beta \}$ as in the proof of Theorem 2.2.

$(1)\ q=121, f(x)=x^2 + 7x + 2$

$(2)\ q=193,\xi =5$

$(3)\ q=289, f(x)=x^2 + 16x + 3$

$(4)\ q=337, \xi =10$

Appendix 2

We list two starter parallel classes in the proof of Lemma 3.2 meeting all the conditions in Lemma 2.3.

$(1)\ q=109, \xi =6$

$(2)\ q=157, \xi =5$

$(3)\ q=181, \xi =2$

$(4)\ q=229, \xi =6$

$(5)\ q=277, \xi =5$

Appendix 3

We list all the base blocks of $(378,3,\{2,3\},2)$ disjoint difference family over $\mathbb {Z}_{378}$ relative to $H=\{0,126,252\}$ in Lemma 3.3. Take $q=379$ and $\xi =2$ in Lemma 3.3 to check the property (3) of Lemma 2.4. Note that in the base block set the three pairs $\{x_i,y_i\}$ $(i=1,2,3)$ are listed in order; they give rise to three blocks $\{\infty _1,\xi ^{x_1},\xi ^{y_1}\},$ $\{\infty _2,\xi ^{x_2},\xi ^{y_2}\}$ and $\{0,\xi ^{x_3},\xi ^{y_3}\}$, as in the proof of [17, Theorem 3.2].