In the study of packet-level random linear network coding (RLNC) in wireless broadcast, RLNC over GF(2^L) is known to asymptotically achieve the optimal completion delay with increasing L. Utilization of guessing random additive noise decoding (GRAND) at physical layer can help leverage RLNC packets to generate syndromes so as to reduce packet erasure probabilities and thus further improve the completion delay performance. Prior to this work, only few studies investigated GRAND-assisted RLNC and they restricted to GF(2)-coding. In this paper, we first provide a general framework to formulate the decoding process of GRAND-assisted RLNC over GF(2^L) for L ≥ 1. Even for GRAND-assisted GF(2)-RLNC, the formulation is more complete than previous considerations in the sense that it takes the a priori information of which packets have errors into consideration. In addition, we propose a novel GRAND-assisted GF(2^L)-RLNC scheme whose computational overhead introduced by GRAND is negligible. We theoretically derive lower bounds on the distribution as well as an upper bound on the expected value of the completion delay of the proposed scheme. Numerical results also demonstrate a reduction in average completion delay for the proposed new GF(2^8)-RLNC scheme, when compared to existing approaches.