## Abstract

The k-fold direct sum encoding of a string a {0; 1}^{n} is a function f_{a} that takes as input sets S [n] of size k and outputs f_{a}(S) = Σ_{iεS} a_{i} (mod 2). In this paper we prove a Direct Sum Testing theorem. We describe a three query test that accepts with probability one any function of the form f_{a} for some a, and rejects with probability (ε) functions f that are "-far from being a direct sum encoding. This theorem has a couple of additional guises: Linearity testing: By identifying the subsets of [n] with vectors in {0; 1}^{n} in the natural way, our result can be thought of as a linearity testing theorem for functions whose domain is restricted to the k'th layer of the hypercube (i.e. the set of n-bit strings with Hamming weight k). Tensor power testing: By moving to -1; 1 notation, the direct sum encoding is equivalent (up to a difference that is negligible when k ≤ √n) to a tensor power. Thus our theorem implies a three query test for deciding if a given tensor f {-1; 1}^{nk} is a tensor power of a single dimensional vector a {-1; 1}^{n}, i.e. whether there is some a such that f = a^{k}. We also provide a four query test for checking if a given ±1 matrix has rank 1. Our test naturally extends the linearity test of Blum, Luby, and Rubinfeld (STOC '90). Our analysis proceeds by first handling the k = n/2 case, and then reducing this case to the general k < n/2 case, using a recent direct product testing theorem of Dinur and Steurer (CCC '2014). The k = n/2 case is proven via a new proof for linearity testing on the hypercube, which we extend to the restricted domain of the n/2-th layer of the hypercube.

Original language | American English |
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Title of host publication | ITCS 2015 - Proceedings of the 6th Innovations in Theoretical Computer Science |

Publisher | Association for Computing Machinery, Inc |

Pages | 327-336 |

Number of pages | 10 |

ISBN (Electronic) | 9781450333337 |

DOIs | |

State | Published - 11 Jan 2015 |

Event | 6th Conference on Innovations in Theoretical Computer Science, ITCS 2015 - Rehovot, Israel Duration: 11 Jan 2015 → 13 Jan 2015 |

### Publication series

Name | ITCS 2015 - Proceedings of the 6th Innovations in Theoretical Computer Science |
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### Conference

Conference | 6th Conference on Innovations in Theoretical Computer Science, ITCS 2015 |
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Country/Territory | Israel |

City | Rehovot |

Period | 11/01/15 → 13/01/15 |

### Bibliographical note

Publisher Copyright:Copyright © 2015 ACM.

## Keywords

- Direct sum
- Linearity testing
- Property testing