## Abstract

Consider a channel W along with a given input distribution PX. In certain settings, such as in the construction of polar codes, the output alphabet of W is 'too large', and hence we replace W by a channel Q having a smaller output alphabet. We say that Q is upgraded with respect to W if W is obtained from Q by processing its output. In this case, the mutual information I(PX,W) between the input and output of W is upper-bounded by the mutual information I(PX,Q) between the input and output of Q. In this paper, we present an algorithm that produces an upgraded channel Q from W, as a function of PX and the required output alphabet size of Q, denoted L. We show that the difference in mutual informations is 'small'. Namely, it is O(L-2/(|X|-1), where |X| is the size of the input alphabet. This power law of L is optimal. We complement our analysis with numerical experiments which show that the developed algorithm improves upon the existing state-of-the-art algorithms also in non-asymptotic setups.

Original language | American English |
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Pages (from-to) | 7822-7836 |

Number of pages | 15 |

Journal | IEEE Transactions on Information Theory |

Volume | 67 |

Issue number | 12 |

DOIs | |

State | Published - 1 Dec 2021 |

### Bibliographical note

Publisher Copyright:© 1963-2012 IEEE.

## Keywords

- Channel upgradation
- channel degradation
- polar codes
- quantization