Building on polar code constructions proposed by
the authors for deterministic broadcast channels, two theorems
are introduced in the present paper for noisy two-user broadcast
channels. The theorems establish polar code constructions for two
important information-theoretic broadcast strategies: (1) Cover’s
superposition strategy; (2) Marton’s construction. One aspect
of the polar code constructions is the alignment of polarization
indices via constraints placed on the auxiliary and channel-input
distributions. The codes achieve capacity-optimal rates for several
classes of broadcast channels (e.g., binary-input stochastically
degraded channels). Applying Arıkan’s original matrix kernel for
polarization, it is shown that the average probability of error in
decoding two private messages at the broadcast receivers decays
asO(2−nβ)where0<β<1 andnisthecodelength.The 2
encoding and decoding complexities remain O(n log n). The error analysis is made possible by defining new polar code ensembles for broadcast channels.