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We consider the problem of communicating a message $m$ in the presence of amalicious jamming adversary (Calvin), who can erase an arbitrary set of up to$pn$ bits, out of $n$ transmitted bits $(x_1,\ldots,x_n)$. The capacity of sucha channel when Calvin is exactly causal, i.e. Calvin's decision of whether ornot to erase bit $x_i$ depends on his observations $(x_1,\ldots,x_i)$ wasrecently characterized to be $1-2p$. In this work we show two (perhaps)surprising phenomena. Firstly, we demonstrate via a novel code constructionthat if Calvin is delayed by even a single bit, i.e. Calvin's decision ofwhether or not to erase bit $x_i$ depends only on $(x_1,\ldots,x_{i-1})$ (andis independent of the "current bit" $x_i$) then the capacity increases to $1-p$when the encoder is allowed to be stochastic. Secondly, we show via a noveljamming strategy for Calvin that, in the single-bit-delay setting, if theencoding is deterministic (i.e. the transmitted codeword is a deterministicfunction of the message $m$) then no rate asymptotically larger than $1-2p$ ispossible with vanishing probability of error, hence stochastic encoding (usingprivate randomness at the encoder) is essential to achieve the capacity of$1-p$ against a one-bit-delayed Calvin.

The benefit of a 1-bit jump-start, and the necessity of stochastic encoding, in jamming channels