This paper considers the problem of broadcasting with side information (BWSI), where a central server broadcasts encoded transmissions using$M$-ary modulation to a set of caching receivers over an additive white Gaussian noise channel. For this problem of noisy index coding with$M$-ary modulated transmission, the ML decoder at any receiver does not involve demodulation to the complex signal point and then index code decoding to the requested message bit. Instead, it decodes directly to the requested message bit, raising the question of whether the central server’s encoding scheme is required to correspond to an index code or if any set of transmissions encoding across the entire library of messages is sufficient. This paper proves that index codes are necessary for solving noisy BWSI problems even with$M$-ary modulated transmission. Further, we look at index-coded QAM for prioritized receivers. An algorithm for mapping index-coded vectors to signal points on a square QAM constellation is presented, which gives mappings that achieve the best ML decoding performance for the prioritized receivers. Expressions for the ML metric seen by the highest priority receiver while using$M$-PSK and$M$-QAM for transmission are derived. For the highest priority receiver,$M$-PSK is shown to outperform$M$-QAM.