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A Matching Vector (MV) family modulo a positive integer m ≥ 2 is a pair of ordered lists U = (u_1, ⋯, u_K) and V = (v_1, ⋯, v_K) where u_i, v_j ∈ ℤ_m^n with the following property: for any i ∈ [K], the inner product ⟨u_i, v_i⟩ = 0 mod m, and for any i ≠ j, ⟨u_i, v_j⟩ ≠ 0 mod m. An MV family is called r-restricted if inner products ⟨u_i, v_j⟩, for all i,j, take at most r different values. The r-restricted MV families are extremely important since the only known construction of constant-query subexponential locally decodable codes (LDCs) are based on them. Such LDCs constructed via matching vector families are called matching vector codes. Let MV(m,n) (respectively MV(m, n, r)) denote the largest K such that there exists an MV family (respectively r-restricted MV family) of size K in ℤ_m^n. Such a MV family can be transformed in a black-box manner to a good r-query locally decodable code taking messages of length K to codewords of length N = m^n. For small prime m, an almost tight bound MV(m,n) ≤ O(m^{n/2}) was first shown by Dvir, Gopalan, Yekhanin (FOCS'10, SICOMP'11), while for general m, the same paper established an upper bound of O(m^{n-1+o_m(1)}), with o_m(1) denoting a function that goes to zero when m grows. For any arbitrary constant r ≥ 3 and composite m, the best upper bound till date on MV(m,n,r) is O(m^{n/2}), is due to Bhowmick, Dvir and Lovett (STOC'13, SICOMP'14).In a breakthrough work, Alrabiah, Guruswami, Kothari and Manohar (STOC'23) implicitly improve this bound for 3-restricted families to MV(m, n, 3) ≤ O(m^{n/3}). In this work, we present an upper bound for r = 3 where MV(m,n,3) ≤ m^{n/6 +O(log n)}, and as a result, any 3-query matching vector code must have codeword length of N ≥ K^{6-o(1)}. 

We study the complexity of lattice problems in a world where algorithms, reductions, and protocols can run in superpolynomial time, revisiting four foundational results: two worst-case to average-case reductions and two protocols. We also show a novel protocol. 1. We prove that secret-key cryptography exists if O˜(n‾√)-approximate SVP is hard for 2εn-time algorithms. I.e., we extend to our setting (Micciancio and Regev's improved version of) Ajtai's celebrated polynomial-time worst-case to average-case reduction from O˜(n)-approximate SVP to SIS. 2. We prove that public-key cryptography exists if O˜(n)-approximate SVP is hard for 2εn-time algorithms. This extends to our setting Regev's celebrated polynomial-time worst-case to average-case reduction from O˜(n1.5)-approximate SVP to LWE. In fact, Regev's reduction is quantum, but ours is classical, generalizing Peikert's polynomial-time classical reduction from O˜(n2)-approximate SVP. 3. We show a 2εn-time coAM protocol for O(1)-approximate CVP, generalizing the celebrated polynomial-time protocol for O(n/logn‾‾‾‾‾‾‾√)-CVP due to Goldreich and Goldwasser. These results show complexity-theoretic barriers to extending the recent line of fine-grained hardness results for CVP and SVP to larger approximation factors. (This result also extends to arbitrary norms.) 4. We show a 2εn-time co-non-deterministic protocol for O(logn‾‾‾‾‾√)-approximate SVP, generalizing the (also celebrated!) polynomial-time protocol for O(n‾√)-CVP due to Aharonov and Regev. 5. We give a novel coMA protocol for O(1)-approximate CVP with a 2εn-time verifier. All of the results described above are special cases of more general theorems that achieve time-approximation factor tradeoffs.