Abstract-Gastpar and Vetterli named the ad-hoc networks with one randomly selected source-destination pair as relay networks and they suggested using arbitrary network coding in order to increase the overall efficiency Here, the relay network is investigated with two relays and no interference and a new achievable rate is obtained using a new network coding (decode-and-broadcast). The obtained achievable rate (i) gives the lower bound for general relay channel, obtained by Cover and El Gamal, but with slight difference due to no interference assumption at the receiver (ii) includes the one relay rates of a two-level relay channel studied by Gupta and Kumar using point to point coding, (iii) includes the rates of two relay Aref network and other special two relay networks, (iv) meets the max-flow min-cut upper bound under certain additional assumptions resulting in certain capacity theorems which include the related previous capacities, (v) is validated by its consistency with previous results relevant to special cases of broadcast channels.

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