Partial-response maximum-likelihood (PRML) methods are now being adopted in many digital magnetic recording systems. It is expected that as linear densities continue to increase, there will be a need to use 'extended' PRML techniques. In fact, commercial systems incorporating extended partial-response target channels, denoted EPRML and EEPRML, employing the EPR4 transfer polynomial h(D) equals 1 plus D minus D2 minus D3 and the EEPR4 transfer polynomial h(D) equals 1 plus 2D minus 2D3 minus D4, respectively, have recently appeared. Among these systems, several apply the rate 2/3, (d,k) equals (1,7) runlength-limited code, originally designed for use with peak-detection, in combination with a detector trellis structure reflecting the d equals 1 constraint. In the EEPR4 case, the d equals 1 constraint is known to provide a coding gain of 2.2 dB, unnormalized for the rate loss, relative to the uncoded channel. In this paper, we describe a nested family of code constraints, properly containing the d equals 1 constraint, intended for use on the EEPR4 channel. These constraints are shown to have the same distance-enhancing properties as the d equals 1 constraint. They permit the design of practical codes for EEPR4 that offer the same coding gain as the (1,7)-coded system, but with higher achievable code rates. The paper concludes with the construction for such a code which, having rate 4/5, offers a 20% increase over the 1,7) code.