In a parallel line assay, there are three treatment
  contrasts of major importance. Block designs allowing the
  estimability of all the three contrasts free from block effects,
  called $L$-designs, necessarily have the block sizes even. For odd
  block sizes, we provide here a class of highly efficient designs,
  called nearly $L$-designs. These nearly $L$-designs have been
  constructed by establishing a link with linear and nearly linear
  trend-free designs.