Consider a positive integer r and a graph G = ( V , E ) with maximum degree Δ and without isolated edges. The least k so that a proper edge colouring c : E → { 1 , 2 , … , k } exists such that ∑ e ∋ u c ( e ) ≠ ∑ e ∋ v c ( e ) for every pair of distinct vertices u , v at distance at most r in G is denoted by χ Σ , r ′ ( G ) . For r = 1 , it has been proved that χ Σ , 1 ′ ( G ) = ( 1 + o ( 1 ) ) Δ . For any r ≥ 2 in turn an infinite family of graphs is known with χ Σ , r ′ ( G ) = Ω ( Δ r − 1 ) . We prove that, on the other hand, χ Σ , r ′ ( G ) = O ( Δ r − 1 ) for r ≥ 2 . In particular, we show that χ Σ , r ′ ( G ) ≤ 6 Δ r − 1 if r ≥ 4 .