On achievability of an ( r , l ) fractional linear network code

It is known that there exists a network, called as the M‐network, which is not scalar linearly solvable but has a vector linear solution for message dimension two. Recently, a generalisation of this result has been presented where it has been shown that for any integer m ≥ 2 , there exists a network which has an ( m , m ) vector linear solution, but does not have a ( w , w ) vector linear solution for w  <  m . This study presents a further generalisation. Specifically, the authors show that for any positive integers k , n , and m ≥ 2 , there exists a network which has a ( mk , mn ) fractional linear solution, but does not have a ( wk , wn ) fractional linear solution for w  <  m .