We study a problem of constructing codes that transform a channel with high bit error rate (BER) into one with low BER (at the expense of rate). Our focus is on obtaining codes with smooth ("graceful") input-output BER curves (as opposed to threshold-like curves typical for long error-correcting codes). To that end we introduce the notion of Low Density Majority Codes (LDMCs). These codes are non-linear sparse-graph codes, which output majority function evaluated on randomly chosen small subsets of the data bits. This is similar to Low Density Generator Matrix codes (LDGMs), except that the XOR function is replaced with the majority. We show that even with a few iterations of belief propagation (BP) the attained input-output curves provably improve upon performance of any linear systematic code. The effect of non-linearity bootstraping the initial iterations of BP, suggests that LDMCs should improve performance in various applications, where LDGMs have been used traditionally (e.g., pre-coding for optics, tornado raptor codes, protograph constructions). As a side result of separate interest we establish a lower (impossibility) bound for the achievable BER of a systematic linear code at one value of erasure noise given its BER at another value. We show that this new bound is superior to the results inferred from the area theorem for EXIT functions.